# 2021-2022 Math Prize Problem #4

• Each month the Mathematics Department offers a math prize problem – a mathematical brain teaser – with a first prize of \$25, a second prize of \$15, and, of course, bragging rights. These problems do not require advanced knowledge of mathematics, just a curious mind and a willingness to slug it out with the problem.

## How many of the numbers 1, 2, 3,…, 5000 are not divisible by any of the numbers 3, 7 and 11?

Submit a solution to this problem to Prof. Ken Ching (kching@mmm.edu, CH 614) by April 20. The first correct submission will win a first prize of 25 dining dollars. All other correct submissions will receive recognition and bragging rights, and one of these submissions will be randomly selected for a second prize of 15 dining dollars. All are welcome to participate, but prizes are for MMC students only. Solution will be posted online and outside CH 603.

## Solution:

Solution to 2021-2022 Math Prize Problem #4: How many of the numbers 1, 2, 3, …, 5000 are not divisible by any of the numbers 3, 7 and 11?

There are 1666 numbers from 1 to 5000 that are divisible by 3 (5000/3 = 1666.6…), 714 numbers divisible by 7 (5000/7 = 714.2…), and 454 numbers divisible by 11 (5000/11 = 454.5…). There are 238 numbers divisible by 3×7 (5000/(3×7) = 238.09…), 151 numbers divisible by 3×11 (5000/(3×11) = 151.5…), 64 numbers divisible by 7×11 (5000/(7×11) = 64.9…) and 21 numbers divisible by 3×7×11 (5000/(3×7×11) = 21.6…). Therefore there are 1666 + 714 + 454 – 238 – 151 – 64 + 21 = 2402 numbers that are divisible by 3, 7 or 11, and hence there are 5000 – 2402 = 2598 numbers that are not divisible by any of the numbers 3, 7 and 11.

First Prize Winner: Juliana Tjornhom, Class of `25.

kching@mmm.edu