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
Set X consists of the positive multiples of 5, and set Y
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- ceilidh.erickson
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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.
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
EdM in Mind, Brain, and Education
Harvard Graduate School of Education