Prime and Composite Numbers - SSAT Upper Level Quantitative

1. If $n$ is a positive integer such that $n^2 + n + 41$ is composite, what is the smallest possible value of $n$?
   Answer: 40 We need to find when $n^2 + n + 41$ is composite. Let's test values systematically. When $n = 40$: $40^2 + 40 + 41 = 1600 + 40 + 41 = 1681 = 41^2$, which is composite. We can verify that for smaller values like $n = 39$: $39^2 + 39 + 41 = 1521 + 39 + 41 = 1601$, which is prime. Choice B (41) gives $41^2 + 41 + 41 = 41(41 + 1 + 1) = 41 \times 43$, which is also composite, but 41 > 40. Choice C (80) gives a composite number but is larger than 40. Choice D (81) also works but is larger.
2. If $p$ is prime, how many positive factors does $p^2$ have?
   Answer: 3 A prime number p has exactly two positive factors: 1 and p. Squaring it adds $p^2$ as a third factor, so $p^2$ has exactly 1, p, and $p^2$. The tempting wrong answer is 2, which forgets to count 1 as a factor.
3. How many prime numbers are there between $2^6$ and $2^6 + 20$, inclusive?
   Answer: 5 First, $2^6 = 64$, so we need primes between 64 and 84, inclusive. Let's check each number systematically: 67 (prime), 71 (prime), 73 (prime), 79 (prime), 83 (prime). All other numbers in this range are composite. For example: 64 = $2^6$, 65 = 5×13, 69 = 3×23, 77 = 7×11, 81 = $3^4$. Therefore, there are exactly 5 prime numbers in this range.
4. If $n$ is the smallest positive integer such that $2n + 1$ is composite, what is the value of $2n + 1$?
   Answer: 9 We need to find the smallest positive integer $n$ such that $2n + 1$ is composite. Let's check values: For $n = 1$: $2(1) + 1 = 3$ (prime). For $n = 2$: $2(2) + 1 = 5$ (prime). For $n = 3$: $2(3) + 1 = 7$ (prime). For $n = 4$: $2(4) + 1 = 9 = 3^2$ (composite). So the smallest value of $n$ is 4, making $2n + 1 = 9$. Choice B (15) corresponds to $n = 7$, choice C (21) to $n = 10$, and choice D (25) to $n = 12$, all of which are larger than $n = 4$.
5. The sum of two prime numbers is 30. If one of the primes is greater than 20, how many such pairs of primes are possible?
   Answer: 1 Let the two primes be $p$ and $q$ where $p + q = 30$ and one prime is greater than 20. The primes greater than 20 and less than 30 are: 23, 29. If $p = 23$, then $q = 30 - 23 = 7$, and 7 is prime. If $p = 29$, then $q = 30 - 29 = 1$, but 1 is not considered prime. Therefore, there is only one pair: (23, 7). Choice B would suggest there are 2 pairs, but the second possibility doesn't work. Choices C and D overcount non-existent pairs.
6. If a and b are different primes and $a + b = 21$, what is $ab$?
   Answer: 38 Since 21 is odd, one of the two primes must be 2. That means the other prime is 21 - 2 = 19. Multiplying gives 2 * 19 = 38. The tempting wrong answer is 34, which comes from 2 and 17, but those sum to 19, not 21.
7. Which number is composite?
   Answer: 119 119 = 7 × 17, so it has factors other than 1 and itself, making it composite. The other choices are prime: 127, 131, and 137 have no divisors besides 1 and themselves. 127 is the tempting trap, but it is prime, not composite.
8. A positive integer $k$ has the property that both $k$ and $k + 2$ are prime. If $k > 3$, how many possible values of $k$ are there less than 50?
   Answer: 5 We're looking for twin prime pairs where both $k$ and $k + 2$ are prime, with $3 < k < 50$. The twin prime pairs in this range are: (5,7), (11,13), (17,19), (29,31), and (41,43). Therefore, the possible values of $k$ are: 5, 11, 17, 29, and 41. This gives us exactly 5 possible values.
9. For a prime $n$ greater than 3, which expression can never be prime?
   Answer: $n^2 + 1$ Any prime greater than 3 is odd. Squaring an odd number gives an odd number, so $n^2$ + 1 is even. Since n > 3, $n^2$ + 1 is greater than 2, so it cannot be prime. The tempting wrong choice is 2n + 1, but it can be prime; for example, n = 5 gives 11.
10. Which of the following statements about the number 91 is true?
    Answer: 91 is composite because it equals 7 × 13 To determine if 91 is prime or composite, we need to check for factors. $91 = 7 \times 13$, so 91 is composite. Choice A is incorrect because checking divisibility by 2, 3, 5, 7 is insufficient - we need to check all primes up to $\sqrt{91} \approx 9.5$, and 91 is divisible by 7. Choice C is wrong because 91 IS divisible by 7 (and 7 is between 2 and 9). Choice D is incorrect because 91 is not divisible by 3 (sum of digits: 9 + 1 = 10, which is not divisible by 3). The correct answer is B because $7 \times 13 = 91$.
11. A number $N$ has exactly 6 positive divisors. If $N = 2^a \cdot 3^b$ where $a$ and $b$ are non-negative integers, and $N > 12$, what is the smallest possible value of $N$?
    Answer: 18 If $N = 2^a \cdot 3^b$, then the number of divisors is $(a+1)(b+1) = 6$. The factor pairs of 6 are (1,6), (2,3), (3,2), and (6,1), giving us $(a,b) = (0,5), (1,2), (2,1), (5,0)$. The corresponding values are: $N = 3^5 = 243$, $N = 2^1 \cdot 3^2 = 18$, $N = 2^2 \cdot 3^1 = 12$, and $N = 2^5 = 32$. Since $N > 12$, we exclude $N = 12$. Among the remaining candidates (18, 32, 243), the smallest is 18.
12. If $p$ and $q$ are distinct primes, which statement must be true?
    Answer: $pq$ is composite. Since p and q are distinct primes, pq has four distinct positive divisors: 1, p, q, and pq. That makes pq composite, not prime. The tempting wrong choice is p times q is odd, but that fails when one of the primes is 2, because 2 times an odd prime is even.
13. If $m = 2^3 \cdot 3^2 \cdot 5$ and $n = 2^2 \cdot 3 \cdot 7$, what can be concluded about $m + n$?
    Answer: $m + n$ is composite because $m + n = 444$, which has multiple factors First, let's calculate $m$ and $n$: $m = 2^3 \cdot 3^2 \cdot 5 = 8 \cdot 9 \cdot 5 = 360$ and $n = 2^2 \cdot 3 \cdot 7 = 4 \cdot 3 \cdot 7 = 84$. Therefore, $m + n = 360 + 84 = 444$. Since $444 = 4 \cdot 111 = 4 \cdot 3 \cdot 37 = 12 \cdot 37$, we see that 444 is composite. Choice A is wrong because we have specific values, not a general statement. Choice B is incorrect because the sum of composite numbers isn't necessarily composite. Choice D is wrong because we can determine the exact nature of this specific sum.

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