2001 AMC 10 Problem 12

Problem 12 of 25IntermediateNumber Theory Logic

Suppose that

n

is the product of three consecutive integers and that

n

is divisible by

7.

Which of the following is not necessarily a divisor of

n?

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Solution

Among three consecutive integers, at least one is even and one is a multiple of

3,

n

is divisible by

6.

With the given factor of

7,

it is divisible by

6,

14,

21,

and

42.

But

28=2^2\cdot7

requires two factors of

2,

which is not guaranteed:

5\cdot6\cdot7=210

is divisible by

7

but not by

28.

Thus, the correct answer is D.

Tagged: divisibility · prime factorization · counterexample