1.How many pairs of positive integers (m, n) satisfy
where n is an odd integer less than 60?
Choose one answer.
a. 5
b. 3
c. 4
d. 6
e. 7
How many pairs of positive integers m, n satisfy
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This problem seems beyond the scope of the GMAT.
That said, here is one approach:
1/m = 1/12 - 4/n.
Since m must be positive, 1/12 must be greater than 4/n:
1/12 > 4/n
n > 48.
Since n must be an odd integer less than 60, test odd integers between 48 and 60 to see which yield integer values for m.
n=49:
1/m = 1/12 - 4/49
1/m = 1/(12*49)
Here, m = 12*49.
n=51:
1/m = 1/12 - 4/51
1/m = 3/(12*51)
1/m = 1/(4*51).
Here, m = 4*51.
n=53:
1/m = 1/12 - 4/53
1/m = 5/(12*53).
Here, m is not an integer.
n=55
1/m = 1/12 - 4/55
1/m = 7/(12*55).
Here, m is not an integer.
n=57:
1/m = 1/12 - 4/57
1/m = 9/(12*57)
1/m = 1/(3*19).
Here, m = 3*19.
n=59:
1/m = 1/12 - 4/59
1/m = 11/(12*59).
Here, m is not an integer.
Only the 3 options in red are viable.
The correct answer is B.
That said, here is one approach:
1/m = 1/12 - 4/n.
Since m must be positive, 1/12 must be greater than 4/n:
1/12 > 4/n
n > 48.
Since n must be an odd integer less than 60, test odd integers between 48 and 60 to see which yield integer values for m.
n=49:
1/m = 1/12 - 4/49
1/m = 1/(12*49)
Here, m = 12*49.
n=51:
1/m = 1/12 - 4/51
1/m = 3/(12*51)
1/m = 1/(4*51).
Here, m = 4*51.
n=53:
1/m = 1/12 - 4/53
1/m = 5/(12*53).
Here, m is not an integer.
n=55
1/m = 1/12 - 4/55
1/m = 7/(12*55).
Here, m is not an integer.
n=57:
1/m = 1/12 - 4/57
1/m = 9/(12*57)
1/m = 1/(3*19).
Here, m = 3*19.
n=59:
1/m = 1/12 - 4/59
1/m = 11/(12*59).
Here, m is not an integer.
Only the 3 options in red are viable.
The correct answer is B.
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Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
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