HMMT 十一月 2019 · GEN 赛 · 第 2 题

HMMT November 2019 — GEN Round — Problem 2

Meghana writes two (not necessarily distinct) primes q and r in base 10 next to each other on a blackboard, resulting in the concatenation of q and r (for example, if q = 13 and r = 5, the number on the blackboard is now 135). She notices that three more than the resulting number is the square of a prime p . Find all possible values of p .

解析

Meghana writes two (not necessarily distinct) primes q and r in base 10 next to each other on a blackboard, resulting in the concatenation of q and r (for example, if q = 13 and r = 5, the number on the blackboard is now 135). She notices that three more than the resulting number is the square of a prime p . Find all possible values of p . Proposed by: Carl Schildkraut Answer: 5 2 Trying p = 2, we see that p − 3 = 1 is not the concatenation of two primes, so p must be odd. Then 2 p − 3 is even. Since r is prime and determines the units digit of the concatenation of q and r , r must 2 be 2. Then p will have units digit 5, which means that p will have units digit 5. Since p is prime, we 2 find that p can only be 5, and in this case, p − 3 = 22 allows us to set q = r = 2 to satisfy the problem statement. So there is a valid solution when p = 5, and this is the only possibility.