CEMC Summer Problem Solving Course
2022
Assignment #5: Due August 5 at 4:00 PM EDT
1. What is the largest integer k for which 3k divides 100! ?
2. Find all pairs (a, b) of non negative integers that satisfy the equation ab − 4a + 3b = 36.
3. How many pairs (x, y) of integers satisfy 18x + 30y = 2022 and xy > 1450?
4. The integer n = 3A4BC where A, B, and C are the thousands, tens, and units digits, respectively. Find all possible values of n for which n is a multiple of 220.
5. What is the smallest positive integer with exactly 2022 positive divisors?
6. Ian and Mike are playing a game where Mike randomly selects a positive three digit integer
and asks Ian whether or not the integer is prime. Ian is lazy, so rather than actually checking
to see if the integer is prime, he does the following.
• He checks to see if the integer is divisible by 2, 3, or 5 since this is easy.
• If he finds that the integer is divisible by at least one of 2, 3, or 5, he says that the integer
is not prime.
• Otherwise, he says that the integer is prime.
Given that Ian never makes a mistake checking for divisibility by 2, 3, or 5 and that there are
exactly 143 three digit prime numbers, what is the probability that Ian is correct?
7. Show that the 6 digit integer ABCDEF (this is not a product, A through F are the digits) is
a multiple of 7 if and only if 5A + 4B + 6C + 2D + 3E + F is also a multiple of 7. [To prove this
“if and only if” statement, you need to prove two things: “if ABCDEF is a multiple of 7, then
5A + 4B + 6C + 2D + 3E + F is a multiple of 7”, as well as “if 5A + 4B + 6C + 2D + 3E + F
is a multiple of 7, then ABCDEF is a multiple of 7”.]
8. We will call the integer n doubly square if n is a perfect square and its number of positive
divisors is a perfect square. What is the smallest doubly square integer that is greater than
2022?