The integer K is positive, but less than 400.

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M7MBA: Moderator; Posts: 2058; Joined: Sun Oct 29, 2017 4:24 am; Thanked: 1 times; Followed by:5 members

The integer K is positive, but less than 400.

by M7MBA » Sun Nov 19, 2017 8:33 am

The integer K is positive, but less than 400. If 21K is a multiple of 180, how many unique prime factors does K have?

A. 1
B. 2
C. 3
D. 4
E. 5

The OA is C.

Experts, may you help me here? I don't have idea how could I solve this PS question. Thanks in advance.

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Source: Beat The GMAT — Problem Solving | Sun Nov 19, 2017 8:33 am

DavidG@VeritasPrep: Legendary Member; Posts: 2663; Joined: Wed Jan 14, 2015 8:25 am; Location: Boston, MA; Thanked: 1153 times; Followed by:128 members; GMAT Score:770

by DavidG@VeritasPrep » Sun Nov 19, 2017 9:42 am

M7MBA wrote:The integer K is positive, but less than 400. If 21K is a multiple of 180, how many unique prime factors does K have?

A. 1
B. 2
C. 3
D. 4
E. 5

The OA is C.

Experts, may you help me here? I don't have idea how could I solve this PS question. Thanks in advance.

Pick the easiest possible number, If K = 180, clearly 12*180 is a multiple of 180.

180 = $$2^2 * 3^2 * 5$$

Prime bases: 2, 3, and 5. The answer is C

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8086; Joined: Sat Apr 25, 2015 10:56 am; Location: Los Angeles, CA; Thanked: 43 times; Followed by:29 members

by Scott@TargetTestPrep » Wed Oct 16, 2019 6:07 pm

M7MBA wrote:The integer K is positive, but less than 400. If 21K is a multiple of 180, how many unique prime factors does K have?

A. 1
B. 2
C. 3
D. 4
E. 5

The OA is C.

Experts, may you help me here? I don't have idea how could I solve this PS question. Thanks in advance.

Since 180 = 18 x 10 = 3^2 x 2^2 x 5 and 21K = 3 x 7 x K, we see that K must be a multiple of 3 x 2^2 x 5.

In other words, K = 3 x 2^2 x 5 x n = 60n for for some positive integer n. We see that K already has 3 unique prime factors, namely, 2, 3 and 5. However, since K is less than 400, we see that n can't be more than 6. Because n is no more than 6, we see that K can't have any more prime factors other than 2, 3 and 5.

Answer: C

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