When 732 is divided by a positive integer x, the remainder is 12. How many…

Concept

When a number N is divided by x, leaving remainder r, then x must exactly divide (N − r), and x must be strictly greater than r (a remainder is always smaller than its divisor). Also, if N is written in prime-factorized form as ap × bq × cr, the total number of factors of N is (p+1)(q+1)(r+1).

Application

1. Since dividing 732 by x leaves remainder 12, x must exactly divide 732 − 12 = 720, and x must be greater than 12.

2. Prime factorize 720: 720 = 24 × 32 × 51.

3. Total number of factors of 720 = (4 + 1) × (2 + 1) × (1 + 1) = 5 × 3 × 2 = 30.

4. Factors of 720 that are 12 or less: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12 — a total of 10 factors.

5. Since x must be greater than 12, exclude these 10 factors from the 30 total factors: 30 − 10 = 20.

Cross-check

Take a factor of 720 greater than 12, say x = 15: 732 ÷ 15 = 48 remainder 12 — valid, confirming the condition. Take x = 12 itself: 732 ÷ 12 = 61 remainder 0, not 12, so 12 is correctly excluded.

So, there are 20 possible values of x.