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Set X consists of the positive multiples of 5, and set Y

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Set X consists of the positive multiples of 5, and set Y

by swerve » Tue Mar 05, 2019 9:29 am

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Set X consists of the positive multiples of 5, and set Y consists of the odd prime numbers less than 20. If set Z consists of every distinct integer less than 100 that is the product of one element from set X and one element from set Y, then set Z consists of how many elements?

A. 12
B. 14
C. 15
D. 16
E. 18

The OA is B

Source: Princeton Review
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by ceilidh.erickson » Tue Mar 05, 2019 9:56 am
There is no real shortcut to this problem; we simply have to count the results.
Set X: [5, 10, 15, 20, 25...]
Set Y: [3, 5, 7, 11, 13, 17, 19]

Since Set Y is finite, start with the elements of Set Y. Start with 3, and list all of the product of 3 * the elements of Set X.
3 * 5 = 15
3 * 10 = 30
3 * 15 = 45
3 * 20 = 60
3 * 25 = 75
3 * 30 = 90
Anything greater than 30 would yield a product greater than 100, so those are the only results for 3. Now try 5:

5 * 5 = 25
5 * 10 = 50
5 * 15 = 75 --> this is a repeat - doesn't count
5 * 20 = 100 --> we're only looking for less than 100, so this doesn't count.

7 * 5 = 35
7 * 10 = 70

11 * 5 = 55
13 * 5 = 65
17 * 5 = 85
19 * 5 = 95

There are 14 distinct elements that are products of a member of Set X and a member of Set Y.

The answer is B.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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