If 123,456=123a+87 and 234,567=123b+6, how many multiples o

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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Mon May 21, 2018 2:42 am

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[GMAT math practice question]

If 123,456=123a+87 and 234,567=123b+6, how many multiples of 123 lie between 123,456 and 234,567?

A. a
B. b
C. a+b
D. a-b
E. b-a

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Source: Beat The GMAT — Problem Solving | Mon May 21, 2018 2:42 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon May 21, 2018 3:49 am

Max@Math Revolution wrote:[GMAT math practice question]

If 123,456=123a+87 and 234,567=123b+6, how many multiples of 123 lie between 123,456 and 234,567?

A. a
B. b
C. a+b
D. a-b
E. b-a

For any EVENLY SPACED SET:
Count = (biggest - smallest)/increment + 1.
The INCREMENT is the difference between successive terms.

Smallest multiple of 123 between 123a + 87 and 123b + 6:
123a +123.
Biggest multiple of 123 between 123a + 87 and 123b + 6:
123b.
Since 123 is the increment between successive multiples of 123, we get:
Count = [(123b) - (123a+123)]/123 + 1 = (123b - 123a - 123)/123 + 1 = (b - a - 1) + 1 = b-a.

The correct answer is E.

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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Wed May 23, 2018 5:03 pm

=>

Since 123,456 = 123*1003 + 87, we must have a = 1003.
Since 234,567 = 123*1907 + 6, we must have b = 1907.
The multiples of 123 between 123,456 and 234,567 are 123*1004, 123*1005, …, 123*1907.
Thus, 1907 – 1004 + 1 = 1907 – 1003 = b – a multiples of 123 lie between 123,456 and 234,567.

Therefore, E is the answer.
Answer E

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