Which of the following numbers will completely divide (729 - 730 - 731 + 732) ?

Concept

When terms contain powers of the same base, factor out the smallest common power. The remaining bracket is then an ordinary integer expression.

A candidate divides the product exactly when its prime factors, with the required multiplicities, are all present in that product.

Application

1. Factor out the smallest power: 729 − 730 − 731 + 732 = 729(1 − 7 − 72 + 73).

2. Evaluate the bracket: 1 − 7 − 49 + 343 = 288. Hence the expression equals 729 × 288.

3. Prime-factorize the integer coefficient: 288 = 25 × 32. Thus the whole product is 25 × 32 × 729.

4. Compare the offered numbers by their prime factorizations.

Cross-check

Dividing by 72 gives (729 × 288) ÷ 72 = 4 × 729, an integer. The other offered numbers require a factor 13 or 17, neither of which occurs in 25 × 32 × 729.

Result

Therefore, 72 completely divides the given expression.