Divisibility Rules - SSAT Middle Level Quantitative

Which of the following statements about divisibility is always true?
Answer: If a number is divisible by 2 and 3, then it is divisible by 6 When you encounter divisibility questions, you need to understand how factors and multiples relate to each other. The key principle is that if a number is divisible by two factors, it's divisible by their least common multiple (LCM), but not necessarily by their product. Let's examine why answer C is correct. If a number is divisible by both 2 and 3, then it must contain both 2 and 3 as factors. Since 2 and 3 are relatively prime (they share no common factors), their LCM is simply $2 \times 3 = 6$ . Therefore, any number divisible by both 2 and 3 is always divisible by 6. Now for the incorrect choices: Answer A fails because being divisible by 6 doesn't guarantee divisibility by 12. For example, 18 is divisible by 6 but not by 12. Answer B is wrong because divisibility by 4 and 6 doesn't always mean divisibility by 24. The LCM of 4 and 6 is 12, not 24, so numbers like 12 are divisible by both 4 and 6 but not by 24. Answer D is incorrect because divisibility by 8 doesn't guarantee divisibility by 16. Consider 24: it's divisible by 8 but not by 16. Remember this pattern: when a number is divisible by two factors, it's guaranteed to be divisible by their LCM, not their product. Always find the LCM when combining divisibility conditions, and test the "always true" statements with small counterexamples to eliminate wrong answers quickly.
Which of the following numbers is divisible by 10?
Answer: 4,120 This question tests middle school number properties and integer skills, specifically the understanding and application of divisibility rules. Divisibility rules help determine if one number divides another without a remainder using specific patterns. For example, a number is divisible by 10 if it ends in 0. In this case, students apply the rule for 10. Choice B is correct because 4,120 ends in 0. This demonstrates the student's ability to recognize and apply the rule accurately. Choice A is incorrect because 7,215 ends in 5, not 0. To help students: Focus on teaching each rule with examples, practice identifying patterns in numbers, and encourage students to verify their answers by applying the rule. Watch for common errors like confusing 10 with 5.
Which of the following numbers is divisible by 2?
Answer: 842 This question tests middle school number properties and integer skills, specifically the understanding and application of divisibility rules. Divisibility rules help determine if one number divides another without a remainder using specific patterns. For example, a number is divisible by 2 if its last digit is even. In this case, students apply the rule that a number is divisible by 2 if it ends in 0, 2, 4, 6, or 8. Choice B is correct because 842 ends in 2, which is even, confirming it is divisible by 2. This demonstrates the student's ability to recognize and apply the rule accurately. Choice A is incorrect because 317 ends in 7, which is odd. To help students: Focus on teaching each rule with examples, practice identifying patterns in numbers, and encourage students to verify their answers by applying the rule. Watch for common errors like confusing even and odd digits.
The sum of the digits of a two-digit number is 12. If the number is divisible by 3 but not by 9, and is a multiple of 8, which of the following could be the number?
Answer: 48 Check each condition in turn. Every choice has a digit sum of 12: $4+8$ , $5+7$ , $6+6$ , $7+5$ and $8+4$ . Because 12 is divisible by 3 but not by 9, every choice is divisible by 3 and none is divisible by 9, so those two conditions do not narrow the list. The deciding condition is being a multiple of 8. Since $48 = 8 \times 6$ , 48 works. Next, $84 \div 8 = 10.5$ , so 84 fails even though it uses the same digits as 48. The numbers 57 and 75 are odd, so they cannot be multiples of 8, and $66 \div 8 = 8.25$ .
The number $2^a \times 3^b \times 5^c$ , where $a$ , $b$ , and $c$ are nonnegative integers, is divisible by 72. What is the smallest possible value of $a + b + c$ ?
Answer: 5 Start with the prime factorization of 72: $72 = 8 \times 9 = 2^3 \times 3^2$ . For $2^a \times 3^b \times 5^c$ to be divisible by 72, it needs at least three factors of 2 and two factors of 3, so $a \geq 3$ and $b \geq 2$ . 72 has no factor of 5, so $c$ has no minimum. Because $c$ may be any nonnegative integer, the smallest choice is $c = 0$ . The smallest possible sum is $a + b + c = 3 + 2 + 0 = 5$ . A sum of 6 comes from assuming $c$ must be at least 1 because 5 appears in the expression, but the stem allows $c = 0$ . Larger sums such as 7 or 8 use more factors of 2 or 3 than divisibility by 72 requires, for example $a = 3, b = 3, c = 1$ . Strategy tip: Factor the divisor into primes first, then use exactly the minimum power of each prime it needs. Primes the divisor lacks can have exponent 0.
Identify the number that is NOT divisible by 4.
Answer: 154 This question tests middle school number properties and integer skills, specifically the understanding and application of divisibility rules. Divisibility rules help determine if one number divides another without a remainder using specific patterns. For example, a number is divisible by 4 if its last two digits form a number divisible by 4. In this case, students check each option using the rule for 4. Choice C is correct because 154's last two digits are 54, and 54 divided by 4 is 13.5, not an integer. This demonstrates the student's ability to recognize and apply the rule accurately. Choice A is incorrect because 316's last two digits are 16, which is divisible by 4. To help students: Focus on teaching each rule with examples, practice identifying patterns in numbers, and encourage students to verify their answers by applying the rule. Watch for common errors like miscalculating the last two digits.
Which of the following numbers is divisible by 8?
Answer: 1,112 This question tests middle school number properties and integer skills, specifically the understanding and application of divisibility rules. Divisibility rules help determine if one number divides another without a remainder using specific patterns. For example, a number is divisible by 8 if its last three digits form a number divisible by 8. In this case, students apply the rule for 8 to each option. Choice A is correct because 1,112's last three digits are 112, and 112 ÷ 8 = 14. This demonstrates the student's ability to recognize and apply the rule accurately. Choice B is incorrect because 1,114's last three are 114, and 114 ÷ 8 = 14.25. To help students: Focus on teaching each rule with examples, practice identifying patterns in numbers, and encourage students to verify their answers by applying the rule. Watch for common errors like checking only the last digit.
A number $N$ has exactly three prime factors: 2, 3, and 5. If $N$ is divisible by 12 but not by 18, which of the following could be $N$ ?
Answer: $2^2 \times 3^1 \times 5^1$ Since $12 = 2^2 \times 3$ , $N$ needs at least $2^2$ and $3^1$ . Since $18 = 2 \times 3^2$ and $N$ already has a factor of 2, the only way to avoid divisibility by 18 is to have exactly one factor of 3. $2^2 \times 3^1 \times 5^1 = 60$ meets both conditions: $60 ÷ 12 = 5$ , while $60 ÷ 18$ is not a whole number. $2^3 \times 3^2 \times 5^1$ , $2^2 \times 3^2 \times 5^2$ and $2^4 \times 3^2 \times 5^1$ each contain $3^2$ along with a factor of 2, so each is divisible by 18; the extra factors of 2 or 5 do not change that. $2^1 \times 3^1 \times 5^2 = 150$ has only one factor of 2, so it is not divisible by 12. Strategy tip: Turn each divisibility condition into a rule about exponents, then check each choice against every rule.
Which of the following numbers is divisible by 6?
Answer: 204 This question tests middle school number properties and integer skills, specifically the understanding and application of divisibility rules. Divisibility rules help determine if one number divides another without a remainder using specific patterns. For example, a number is divisible by 6 if it is divisible by both 2 and 3. In this case, students check each number for divisibility by 6. Choice B is correct because 204 is even and its digit sum (2+0+4=6) is divisible by 3. This demonstrates the student's ability to recognize and apply the rule accurately. Choice A is incorrect because 123 is odd, so not divisible by 2. To help students: Focus on teaching each rule with examples, practice identifying patterns in numbers, and encourage students to verify their answers by applying the rule. Watch for common errors like forgetting to check both 2 and 3.
Which of the following numbers is divisible by 5?
Answer: 9,640 This question tests middle school number properties and integer skills, specifically the understanding and application of divisibility rules. Divisibility rules help determine if one number divides another without a remainder using specific patterns. For example, a number is divisible by 5 if it ends in 0 or 5. In this case, students check the last digit of each number. Choice C is correct because 9,640 ends in 0. This demonstrates the student's ability to recognize and apply the rule accurately. Choice A is incorrect because 4,318 ends in 8, not 0 or 5. To help students: Focus on teaching each rule with examples, practice identifying patterns in numbers, and encourage students to verify their answers by applying the rule. Watch for common errors like overlooking the last digit.
Identify the number that is NOT divisible by 10.
Answer: 7,001 This question tests middle school number properties and integer skills, specifically the understanding and application of divisibility rules. Divisibility rules help determine if one number divides another without a remainder using specific patterns. For example, a number is divisible by 10 if it ends in 0. In this case, students check the last digit for not ending in 0. Choice C is correct because 7,001 ends in 1, not 0. This demonstrates the student's ability to recognize and apply the rule accurately. Choice A is incorrect because 3,450 ends in 0. To help students: Focus on teaching each rule with examples, practice identifying patterns in numbers, and encourage students to verify their answers by applying the rule. Watch for common errors like confusing with 5.
Identify the number that is NOT divisible by 2.
Answer: 731 This question tests middle school number properties and integer skills, specifically the understanding and application of divisibility rules. Divisibility rules help determine if one number divides another without a remainder using specific patterns. For example, a number is divisible by 2 if its last digit is even. In this case, students check the last digit for oddness to find the one not divisible by 2. Choice B is correct because 731 ends in 1, which is odd. This demonstrates the student's ability to recognize and apply the rule accurately. Choice A is incorrect because 908 ends in 8, even. To help students: Focus on teaching each rule with examples, practice identifying patterns in numbers, and encourage students to verify their answers by applying the rule. Watch for common errors like misidentifying even digits.
Identify the number that is NOT divisible by 3.
Answer: 517 This question tests middle school number properties and integer skills, specifically the understanding and application of divisibility rules. Divisibility rules help determine if one number divides another without a remainder using specific patterns. For example, a number is divisible by 3 if the sum of its digits is divisible by 3. In this case, students check the digit sum for each option. Choice C is correct because 517's digit sum is 5+1+7=13, and 13 is not divisible by 3. This demonstrates the student's ability to recognize and apply the rule accurately. Choice A is incorrect because 222's sum is 6, divisible by 3. To help students: Focus on teaching each rule with examples, practice identifying patterns in numbers, and encourage students to verify their answers by applying the rule. Watch for common errors in summing digits.
Which of the following numbers is divisible by 9?
Answer: 4,563 A number is divisible by 9 exactly when the sum of its digits is divisible by 9. Find each digit sum: - 1,234: 1 + 2 + 3 + 4 = 10, not a multiple of 9. - 2,716: 2 + 7 + 1 + 6 = 16, not a multiple of 9. - 4,563: 4 + 5 + 6 + 3 = 18, a multiple of 9. - 5,432: 5 + 4 + 3 + 2 = 14, not a multiple of 9. Only 4,563 works, and indeed 4,563 ÷ 9 = 507. A common mistake is to use the divisibility rule for 3 instead: a digit sum divisible by 3 is not enough for 9. Another is to check only whether the number is odd or ends in a certain digit; for 9, only the digit sum matters. The number 2,716 is close to a multiple of 9 (2,718), but its digit sum, 16, is not a multiple of 9. To check your choice, divide: if the result is a whole number, the number is divisible by 9.
Which of the following numbers is divisible by 4?
Answer: 1,096 This question tests middle school number properties and integer skills, specifically the understanding and application of divisibility rules. Divisibility rules help determine if one number divides another without a remainder using specific patterns. For example, a number is divisible by 4 if its last two digits are divisible by 4. In this case, students check the last two digits of each. Choice D is correct because 1,096's last two are 96, and 96 ÷ 4 = 24. This demonstrates the student's ability to recognize and apply the rule accurately. Choice A is incorrect because 1,062's last two are 62, and 62 ÷ 4 = 15.5. To help students: Focus on teaching each rule with examples, practice identifying patterns in numbers, and encourage students to verify their answers by applying the rule. Watch for common errors in division of last digits.
How many whole numbers from 1 to 50 are divisible by either 2 or 3, but not by both?
Answer: 25 From 1 to 50 there are 25 multiples of 2, 16 multiples of 3 (since $3 \times 16 = 48$ ), and 8 multiples of both, which are the multiples of 6 (since $6 \times 8 = 48$ ). Divisible by 2 but not 3: $25 - 8 = 17$ . Divisible by 3 but not 2: $16 - 8 = 8$ . Divisible by exactly one of them: $17 + 8 = 25$ . 33 is the count divisible by 2 or 3 with the overlap kept: $25 + 16 - 8 = 33$ . It includes the 8 multiples of 6, which the words "but not by both" exclude, so the 8 must be removed from each group, for $41 - 16 = 25$ . Strategy tip: For "either but not both," subtract the overlap from each set separately, then add the two remainders.
If a number is divisible by 8, what must be true about its digits?
Answer: The last three digits are divisible by 8. This question tests middle school number properties and integer skills, specifically the understanding and application of divisibility rules. Divisibility rules help determine if one number divides another without a remainder using specific patterns. For example, a number is divisible by 8 if its last three digits form a number divisible by 8. In this case, students recall the rule for 8. Choice A is correct because it accurately states the last three digits rule. This demonstrates the student's ability to recognize and apply the rule accurately. Choice B is incorrect because the last digit alone does not determine divisibility by 8. To help students: Focus on teaching each rule with examples, practice identifying patterns in numbers, and encourage students to verify their answers by applying the rule. Watch for common errors like confusing 8 with 2 or 4.

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