Multiple choice

A three-digit number $N$ leaves the same remainder upon dividing $68488$ and $67516$. How many possible values does $N$ have?

  1. $8$
  2. $6$
  3. $5$
  4. $4$
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B Correct answer
Explanation

N must be a divisor of the difference 68488 - 67516 = 972. The prime factorization of 972 is 2^2 * 3^5. The divisors of 972 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 243, 324, 486, 972. Three-digit divisors are 108, 162, 243, 324, 486, 972, which are 6 values.

AI explanation

If a number N leaves the same remainder when dividing 68488 and 67516, N perfectly divides their difference. Subtract the numbers to get 68488 - 67516 = 972. Find the factors of 972 that are three-digit numbers: 108, 162, 243, 324, 486, and 972. There are 6 possible values for N.

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