Official Guide Explanation:
Problem Solving #142




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Solution and Metadata

Question: 142
Page: 172
Difficulty: 6 (Moderately Difficult)
Category 1: Arithmetic > Properties of Integers > Factors and Multiples

Explanation: The question tells us that 3150y is the square of an integer. There are an infinite number of possible values, but we're looking for the smallest one.

To find that, we first need to know the prime factorization of 3150:

3150 = 63(50)

= 7(9)(5)(10)

= 7(3)(3)(5)(2)(5)

= 2 * 32 * 52 * 7

The prime factorization of the square of an integer will contain nothing but even powers. So, when we multiply 3150 by y, the goal is to make all the odd powers even. Since the powers of 3 and 5 are already even, we don't need to change them. However, since the powers of 2 and 7 are each 1, we'll need to multiply the number by an additional 2 and an additional 7:

= (2 * 32 * 52 * 7) * (2 * 7)

We need to include both 2 and 7. By multiplying 3150 by 14, the result is the square of an integer. Choice (E) is correct.

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