Prime and Composite Numbers - SSAT Middle Level Quantitative

Prime numbers have exactly two factors; composite numbers have more than two. Examples: 2, 3, 5, 7 are prime; 4 and 6 are composite. Is 1 prime or composite?
Answer: 1 is neither prime nor composite. This question tests middle-level SSAT skills: determining whether a number is prime or composite, focusing on number properties and integers. Prime numbers have only two distinct positive divisors: 1 and the number itself, while composite numbers have more than two. In this question, students must use the definitions and identify factors, for example, recognizing that 1 has only one divisor, which is itself. Choice C is correct because it accurately reflects that 1 is neither prime nor composite, showing understanding of number classification. Choice A is incorrect because it demonstrates a common misconception, such as assuming 1 is prime due to its oddness. To help students: Encourage practice with factorization and using divisibility rules. Teach them to verify by checking divisibility by known smaller primes (2, 3, 5, 7) and practice spotting common mistakes like assuming number patterns.
The number $91$ can be written as the product of two prime numbers. What is the sum of these two prime numbers?
Answer: $20$ When you encounter a question about writing a number as the product of prime numbers, you're working with prime factorization. This means breaking down a composite number into its prime building blocks. To find the prime factorization of $91$ , start by testing small prime numbers as divisors. Since $91$ is odd, it's not divisible by $2$ . Check if it's divisible by $3$ by adding its digits: $9 + 1 = 10$ , which isn't divisible by $3$ , so $91$ isn't either. Next, try $5$ - since $91$ doesn't end in $0$ or $5$ , it's not divisible by $5$ . Now test $7$ : $91 ÷ 7 = 13$ . Since $13$ is also prime, we have $91 = 7 × 13$ . The sum of these two prime factors is $7 + 13 = 20$ . Looking at the wrong answers: $16$ would come from $3 + 13$ , but $3 × 13 = 39$ , not $91$ . $18$ might result from $5 + 13$ , but $5 × 13 = 65$ . $14$ could come from $7 + 7$ , but $7 × 7 = 49$ . $12$ could come from $5 + 7$ , but $5 × 7 = 35$ . Strategy tip: When finding prime factorization, work systematically through small primes: $2, 3, 5, 7, 11, 13...$ Use divisibility rules to save time - check if the sum of digits is divisible by $3$ , or if the number ends in $0$ or $5$ for divisibility by $5$ .
Prime numbers have exactly two factors; composite numbers have more than two. Examples: 2, 3, 5, 7 are prime; 4 and 6 are composite. Select the prime numbers from the list: 9, 11, 12, 13.
Answer: 11 and 13 This question tests middle-level SSAT skills: determining whether a number is prime or composite, focusing on number properties and integers. Prime numbers have only two distinct positive divisors: 1 and the number itself, while composite numbers have more than two. In this question, students must use the definitions and identify factors, for example, checking that 11 and 13 have only 1 and themselves as divisors. Choice A is correct because it accurately reflects the numbers' properties, showing understanding of prime number identification. Choice B is incorrect because it demonstrates a common misconception, such as failing to recognize that 9 and 12 are composite. To help students: Encourage practice with factorization and using divisibility rules. Teach them to verify by checking divisibility by known smaller primes (2, 3, 5, 7) and practice spotting common mistakes like assuming number patterns.
Prime numbers have exactly two factors; composite numbers have more than two. For examples, 2, 3, 5, 7 are prime; 4 and 6 are composite. How many prime numbers are there between 10 and 20?
Answer: 4 This question tests middle-level SSAT skills: determining whether a number is prime or composite, focusing on number properties and integers. Prime numbers have only two distinct positive divisors: 1 and the number itself, while composite numbers have more than two. In this question, students must use the definitions and identify factors, for example, listing primes like 11, 13, 17, and 19 between 10 and 20. Choice B is correct because it accurately reflects the count of primes in the range, showing understanding of prime identification. Choice A is incorrect because it demonstrates a common misconception, such as undercounting the primes in the interval. To help students: Encourage practice with factorization and using divisibility rules. Teach them to verify by checking divisibility by known smaller primes (2, 3, 5, 7) and practice spotting common mistakes like assuming number patterns.
A prime has exactly two factors; a composite has more than two. In the small set 2-7, 2, 3, 5, 7 are prime and 4, 6 are composite. Which of the following numbers is composite?
Answer: 21 This question tests middle-level SSAT skills: determining whether a number is prime or composite, focusing on number properties and integers. Prime numbers have only two distinct positive divisors: 1 and the number itself, while composite numbers have more than two. In this question, students must use the definitions and identify factors, for example, recognizing that 21 has more than two divisors like 1, 3, 7, and 21. Choice C is correct because it accurately reflects the number's properties, showing understanding of composite number identification. Choice A is incorrect because it demonstrates a common misconception, such as assuming 17 is composite when it is prime. To help students: Encourage practice with factorization and using divisibility rules. Teach them to verify by checking divisibility by known smaller primes (2, 3, 5, 7) and practice spotting common mistakes like assuming number patterns.
Prime numbers have exactly two factors, 1 and itself; composite numbers have more than two. Using 2, 3, 4, 5, 6, and 7 as examples, decide: Is 29 prime or composite?
Answer: 29 is prime because it has two factors. This question tests middle-level SSAT skills: determining whether a number is prime or composite, focusing on number properties and integers. Prime numbers have only two distinct positive divisors: 1 and the number itself, while composite numbers have more than two. In this question, students must use the definitions and identify factors, for example, checking if 29 is prime because its only divisors are 1 and 29. Choice B is correct because it accurately reflects the number's properties, showing understanding of prime number identification. Choice A is incorrect because it demonstrates a common misconception, such as assuming all odd numbers are composite, but oddness does not determine compositeness. To help students: Encourage practice with factorization and using divisibility rules. Teach them to verify by checking divisibility by known smaller primes (2, 3, 5, 7) and practice spotting common mistakes like assuming number patterns.
Prime numbers have exactly two factors; composite numbers have more than two. Examples: 2, 3, 5, 7 are prime; 4 and 6 are composite. Which of the following numbers is composite?
Answer: 57 This question tests middle-level SSAT skills: determining whether a number is prime or composite, focusing on number properties and integers. Prime numbers have only two distinct positive divisors: 1 and the number itself, while composite numbers have more than two. In this question, students must use the definitions and identify factors, for example, recognizing that 57 equals 3 times 19. Choice B is correct because it accurately reflects the number's properties, showing understanding of composite identification. Choice A is incorrect because it demonstrates a common misconception, such as assuming 53 is composite when it is prime. To help students: Encourage practice with factorization and using divisibility rules. Teach them to verify by checking divisibility by known smaller primes (2, 3, 5, 7) and practice spotting common mistakes like assuming number patterns.
What is the product of all the different prime factors of 60?
Answer: 30 When you see a question asking for the product of prime factors, you need to first find the prime factorization of the given number, then multiply each different prime factor exactly once. Let's find the prime factorization of 60. Start by dividing by the smallest prime numbers: $60 = 2^2 \times 3 \times 5 = 4 \times 3 \times 5$ The prime factorization shows us that 60 has three different prime factors: 2, 3, and 5. Notice that 2 appears twice (as $2^2$ ), but we only count each different prime factor once when finding their product. The product of all different prime factors is: $2 \times 3 \times 5 = 30$ Now let's examine why the other answers are incorrect. Answer (B) 60 is the original number itself, not the product of its prime factors—this would be a trap for students who confuse the question. Answer (C) 120 equals $2^3 \times 3 \times 5$ , which might result from incorrectly including an extra factor of 2. Answer (D) 180 equals $2^2 \times 3^2 \times 5$ , which could come from squaring some factors or making calculation errors. The correct answer is (A) 30. Strategy tip: For prime factorization problems, always break the number down completely into prime factors first, identify each unique prime (ignoring how many times each appears), then multiply those unique primes together. Don't let repeated factors confuse you—count each prime only once.
Prime numbers have exactly two factors; composite numbers have more than two. For examples, 2, 3, 5, 7 are prime; 4 and 6 are composite. Which of the following numbers is composite?
Answer: 45 This question tests middle-level SSAT skills: determining whether a number is prime or composite, focusing on number properties and integers. Prime numbers have only two distinct positive divisors: 1 and the number itself, while composite numbers have more than two. In this question, students must use the definitions and identify factors, for example, recognizing that 45 has divisors like 1, 3, 5, 9, 15, and 45. Choice C is correct because it accurately reflects the number's properties, showing understanding of composite identification. Choice A is incorrect because it demonstrates a common misconception, such as assuming 41 is composite when it is prime. To help students: Encourage practice with factorization and using divisibility rules. Teach them to verify by checking divisibility by known smaller primes (2, 3, 5, 7) and practice spotting common mistakes like assuming number patterns.
A prime has exactly two factors; a composite has more than two. In 2–7, 2, 3, 5, 7 are prime while 4 and 6 are composite. Is 2 prime or composite?
Answer: 2 is prime because it has two factors. This question tests middle-level SSAT skills: determining whether a number is prime or composite, focusing on number properties and integers. Prime numbers have only two distinct positive divisors: 1 and the number itself, while composite numbers have more than two. In this question, students must use the definitions and identify factors, for example, noting that 2 has only 1 and 2 as divisors. Choice B is correct because it accurately reflects the number's properties, showing understanding of prime identification. Choice A is incorrect because it demonstrates a common misconception, such as assuming all even numbers are composite, though 2 is an exception. To help students: Encourage practice with factorization and using divisibility rules. Teach them to verify by checking divisibility by known smaller primes (2, 3, 5, 7) and practice spotting common mistakes like assuming number patterns.
A prime has exactly two factors; a composite has more than two. Small examples: 2, 3, 5, 7 are prime; 4 and 6 are composite. Identify the composite numbers in the following set: 18, 19, 20, 21.
Answer: 18, 20, and 21 This question tests middle-level SSAT skills: determining whether a number is prime or composite, focusing on number properties and integers. Prime numbers have only two distinct positive divisors: 1 and the number itself, while composite numbers have more than two. In this question, students must use the definitions and identify factors, for example, noting that 18, 20, and 21 have more than two divisors while 19 does not. Choice A is correct because it accurately reflects the composites in the set, showing understanding of composite identification. Choice D is incorrect because it demonstrates a common misconception, such as assuming only the prime is composite. To help students: Encourage practice with factorization and using divisibility rules. Teach them to verify by checking divisibility by known smaller primes (2, 3, 5, 7) and practice spotting common mistakes like assuming number patterns.
A prime has exactly two factors; a composite has more than two. Small examples: 2, 3, 5, 7 are prime; 4, 6 are composite. Select the prime numbers from the list: 22, 23, 24, 25.
Answer: Only 23 This question tests middle-level SSAT skills: determining whether a number is prime or composite, focusing on number properties and integers. Prime numbers have only two distinct positive divisors: 1 and the number itself, while composite numbers have more than two. In this question, students must use the definitions and identify factors, for example, checking that only 23 has exactly two divisors in the list. Choice A is correct because it accurately reflects the prime in the list, showing understanding of prime identification. Choice B is incorrect because it demonstrates a common misconception, such as including a composite like 22 as prime. To help students: Encourage practice with factorization and using divisibility rules. Teach them to verify by checking divisibility by known smaller primes (2, 3, 5, 7) and practice spotting common mistakes like assuming number patterns.
Prime numbers have exactly two factors; composite numbers have more than two. Examples: 2, 3, 5, 7 are prime; 4 and 6 are composite. What is the smallest prime number greater than 50?
Answer: 53 This question tests middle-level SSAT skills: determining whether a number is prime or composite, focusing on number properties and integers. Prime numbers have only two distinct positive divisors: 1 and the number itself, while composite numbers have more than two. In this question, students must use the definitions and identify factors, for example, checking numbers greater than 50 until finding 53 with only two divisors. Choice C is correct because it accurately reflects the smallest prime greater than 50, showing understanding of prime identification. Choice A is incorrect because it demonstrates a common misconception, such as assuming 51 is prime when it is 3 times 17. To help students: Encourage practice with factorization and using divisibility rules. Teach them to verify by checking divisibility by known smaller primes (2, 3, 5, 7) and practice spotting common mistakes like assuming number patterns.
A student claims that every odd number greater than $1$ is prime. Which of the following is the smallest counterexample to this claim?
Answer: $9$ This question tests your understanding of prime numbers and asks you to find a counterexample—an instance that disproves a general claim. The student's claim is that "every odd number greater than 1 is prime," so you need to find the smallest odd number greater than 1 that is NOT prime. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. Let's check the odd numbers systematically: 3 is prime (factors: 1, 3), 5 is prime (factors: 1, 5), 7 is prime (factors: 1, 7), but 9 is composite because $9 = 3 \times 3$ , giving it factors 1, 3, and 9. Since 9 is the smallest odd number greater than 1 that isn't prime, it's the smallest counterexample to the claim. This makes A) $9$ correct. Looking at the other choices: B) $15$ is indeed composite ( $15 = 3 \times 5$ ), but it's larger than 9, so it's not the smallest counterexample. C) $21$ is also composite ( $21 = 3 \times 7$ ), but again, it's larger than 9. D) $25$ is composite too ( $25 = 5 \times 5$ ), but it's the largest option given. When looking for counterexamples on the SSAT, always work systematically from the smallest possibilities. Don't assume the first few cases represent the general pattern—mathematical claims often have exceptions, and finding the smallest one demonstrates the most efficient logical reasoning.
If $p$ is a prime number greater than $2$ , which of the following expressions must represent a composite number?
Answer: $p^2 - 1$ When you encounter questions about prime numbers and composite numbers, remember that a composite number has factors other than 1 and itself, while a prime has exactly two factors. Since $p$ is a prime greater than 2, it must be odd. Let's examine $p^2 - 1$ , which we can factor as $(p-1)(p+1)$ . Since $p$ is an odd prime greater than 2, both $p-1$ and $p+1$ are even numbers greater than 1. This means $p^2 - 1$ is the product of two integers both greater than 1, making it composite by definition. This factorization works for any odd prime: if $p = 3$ , then $p^2 - 1 = 8 = 2 \times 4$ ; if $p = 5$ , then $p^2 - 1 = 24 = 4 \times 6$ . Choice A ( $p + 2$ ) could be prime—when $p = 3$ , we get $3 + 2 = 5$ , which is prime. Choice C ( $2p + 1$ ) also could be prime—when $p = 3$ , we get $2(3) + 1 = 7$ , which is prime. Choice D ( $p + 4$ ) might be prime too—when $p = 3$ , we get $3 + 4 = 7$ , which is prime. The key insight is that only choice B must always be composite, regardless of which prime $p$ you choose, because it always factors as $(p-1)(p+1)$ . Study tip: When a question asks what "must" be true, look for algebraic expressions you can factor. Factoring often reveals why an expression will always have the required property.
How many prime numbers are there between $40$ and $50$ , inclusive?
Answer: $3$ When you encounter a question asking for prime numbers in a specific range, you need to systematically check each number for primality. A prime number has exactly two factors: 1 and itself. Let's examine each number from 40 to 50: - 40: Even number, divisible by 2, 4, 5, 8, 10, 20 → not prime - 41: Check divisibility by primes up to $\sqrt{41} \approx 6.4$ . Test 2, 3, 5: 41 is odd, $4+1=5$ (not divisible by 3), doesn't end in 0 or 5 → prime - 42: Even number, obviously composite → not prime - 43: Check divisibility by 2, 3, 5: 43 is odd, $4+3=7$ (not divisible by 3), doesn't end in 0 or 5 → prime - 44: Even number → not prime - 45: Ends in 5, divisible by 5 → not prime - 46: Even number → not prime - 47: Check divisibility by 2, 3, 5, 7: 47 is odd, $4+7=11$ (not divisible by 3), doesn't end in 0 or 5, and $47 ÷ 7 = 6.7...$ → prime - 48: Even number → not prime - 49: $7 \times 7 = 49$ → not prime - 50: Even number → not prime The prime numbers are 41, 43, and 47, giving us 3 primes total. Answer choice (A) 1 might come from only finding one obvious prime. Answer choice (B) 2 suggests missing one of the three primes. Answer choice (D) 4 likely includes 49, forgetting that $49 = 7^2$ . Study tip: When checking for primes, you only need to test divisibility by primes up to the square root of your number. This saves time and reduces errors.
A prime has exactly two factors; a composite has more than two. In the examples 2–7, 2, 3, 5, 7 are prime while 4 and 6 are composite. Is 51 prime or composite?
Answer: 51 is composite because $3\times17=51$ . This question tests middle-level SSAT skills: determining whether a number is prime or composite, focusing on number properties and integers. Prime numbers have only two distinct positive divisors: 1 and the number itself, while composite numbers have more than two. In this question, students must use the definitions and identify factors, for example, noting that 51 equals 3 times 17. Choice B is correct because it accurately reflects the number's properties, showing understanding of composite identification. Choice A is incorrect because it demonstrates a common misconception, such as assuming all odd numbers are prime. To help students: Encourage practice with factorization and using divisibility rules. Teach them to verify by checking divisibility by known smaller primes (2, 3, 5, 7) and practice spotting common mistakes like assuming number patterns.
Which of the following statements about the number $1$ is correct?
Answer: $1$ is neither prime nor composite When you encounter questions about classifying numbers, you need to understand the precise definitions of prime and composite numbers. A prime number has exactly two positive divisors: $1$ and itself. A composite number has more than two positive divisors. The number $1$ is a special case that doesn't fit either category. Since $1$ has only one positive divisor (itself), it fails to meet the "exactly two divisors" requirement for prime numbers. Similarly, since $1$ doesn't have more than two divisors, it's not composite either. By mathematical convention, $1$ is classified as neither prime nor composite, making answer choice C correct. Let's examine why the other options are incorrect. Choice A states that $1$ is prime because it has exactly one divisor, but this misunderstands the definition—prime numbers must have exactly two positive divisors, not one. Choice B claims $1$ is composite because it can be written as $1 × 1$ , but being expressible as a product doesn't automatically make a number composite; the key is having more than two divisors. Choice D suggests $1$ is both prime and composite, which is impossible since these categories are mutually exclusive by definition. Remember that $1$ is the only positive integer that's neither prime nor composite—it's in a category by itself. On standardized tests, questions about $1$ often try to trick you by testing whether you know the precise definitions rather than relying on intuition.
A number $N$ is the product of three consecutive positive integers. If the middle integer is prime, what can be concluded about $N$ ?
Answer: $N$ must be composite Write $N = (p-1) \cdot p \cdot (p+1)$ , where the middle integer $p$ is prime. The smallest case is $p = 2$ : $N = 1 \cdot 2 \cdot 3 = 6$ , which is composite. For any larger prime, $N$ has $p$ as a factor that is neither 1 nor $N$ itself; for example, $4 \cdot 5 \cdot 6 = 120$ . So $N$ must be composite and can never be prime, which also rules out the claim that it could be either. Divisibility by 5 is not guaranteed: $1 \cdot 2 \cdot 3 = 6$ and $2 \cdot 3 \cdot 4 = 24$ are not multiples of 5. The claim that $N$ is even but not divisible by 4 holds for 6 but fails for $24$ and $120$ .
A prime has exactly two factors; a composite has more than two. In 2–7, 2, 3, 5, 7 are prime while 4 and 6 are composite. Is 49 prime or composite?
Answer: 49 is composite because $7\times7=49$ . This question tests middle-level SSAT skills: determining whether a number is prime or composite, focusing on number properties and integers. Prime numbers have only two distinct positive divisors: 1 and the number itself, while composite numbers have more than two. In this question, students must use the definitions and identify factors, for example, noting that 49 equals 7 times 7. Choice B is correct because it accurately reflects the number's properties, showing understanding of composite identification. Choice A is incorrect because it demonstrates a common misconception, such as assuming all odd numbers are prime. To help students: Encourage practice with factorization and using divisibility rules. Teach them to verify by checking divisibility by known smaller primes (2, 3, 5, 7) and practice spotting common mistakes like assuming number patterns.

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