The positive integer divisors of 175, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 7?
To solve this problem, I need to find all the positive integer divisors of 175, except 1, and figure out how to arrange them around a circle so that every pair of adjacent integers has a common factor greater than 1. I can start by finding the prime factorization of 175, which is 5^2 * 7. This means that the positive integer divisors of 175, except 1, are 5, 7, 25, 35, and 175. I notice that 7 is the only divisor that is not a multiple of 5, so it must be adjacent to 35 and 175, which are both multiples of 7. This leaves 5 and 25 to be adjacent to each other, and to 35 and 175, respectively. So one possible arrangement of the divisors around a circle is 5, 35, 7, 175, 25. The sum of the two integers adjacent to 7 is 35 + 175, which is 210. 