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If x and y are integers, what is the least possible positive

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by Max@Math Revolution » Mon Sep 03, 2018 11:20 pm

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[Math Revolution GMAT math practice question]

If x and y are integers, what is the least possible positive value of 21x+35y?

A. 1
B. 3
C. 5
D. 7
E. 9
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by fskilnik@GMATH » Tue Sep 04, 2018 12:19 pm
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]
If x and y are integers, what is the least possible positive value of 21x+35y?
A. 1
B. 3
C. 5
D. 7
E. 9
Nice problem, in which STRUCTURE does the most part of it, as we see below!
\[x,y\,\,{\text{ints}}\,\,\left( * \right)\]
\[{\text{?}}\,\,{\text{ = }}\,\,{\left( {21x + 35y} \right)_{\,\min \, > \,0}} = \,\,\,{\left[ {7\left( {3x + 5y} \right)} \right]_{\,\min \, > \,0}}\,\,\,\mathop = \limits^{\left( * \right)} \,\,\,{\left[ {7\left( {3x + 5y} \right)} \right]_{\,\min \, \geqslant \,1}}\]
\[{?_{{\text{temporary}}}}\,\, = \,\,{\left[ {3x + 5y\,\,\mathop = \limits^{\left( * \right)} \,\,\operatorname{int} } \right]_{\,\min \, \geqslant \,1}} \geqslant 1\]
\[{\text{Take}}\,\,\left( {x,y} \right) = \left( { - 3,2} \right)\,\,\, \Rightarrow 3x + 5y = 1 = {\left[ {3x + 5y} \right]_{\,\min \, \geqslant \,1}}\]
\[? = 7\]

This solution follows the notations and rationale taught in the GMATH method.

Regards,
fskilnik.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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by GMATGuruNY » Tue Sep 04, 2018 12:48 pm
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If x and y are integers, what is the least possible positive value of 21x+35y?

A. 1
B. 3
C. 5
D. 7
E. 9
Make a list of integer options for |21x| and |35y|:
|21x| = 0, 21, 42, 63, 84, 105...
|35y| = 0, 35, 70, 105...

The smallest positive sum is yielded in the following cases:
21x = 42 and 35y = -35, with the result that 21x + 35y = 42 - 35 = 7
21x = -63 and 35y = 70, with the result that 21x + 35y = -63 + 70 = 7
Even if we extend the two lists, a smaller positive sum is not possible.

The correct answer is D.
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by Brent@GMATPrepNow » Tue Sep 04, 2018 12:53 pm
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If x and y are integers, what is the least possible positive value of 21x + 35y?

A. 1
B. 3
C. 5
D. 7
E. 9
21x + 35y = 7(3x + 5y)

Since x and y are INTEGERS, we know that 3x and 5y are each INTEGERS, which means 3x + 5y equals some integer
So, we can write: 21x + 35y = 7(some integer)
In other words, 21x + 35y is a multiple of 7

Check the answer choices.....only D is a multiple of 7

Answer: D

Cheers,
Brent
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by Brent@GMATPrepNow » Tue Sep 04, 2018 1:02 pm
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If x and y are integers, what is the least possible positive value of 21x+35y?

A. 1
B. 3
C. 5
D. 7
E. 9
Another approach:

There's a nice rule that says: If j, k and d are integers, and j is a multiple of d, and k is a multiple of d, then (j+k) must be a multiple of d

21x is a multiple of 7 (since 21x = 7(3x)
35y is a multiple of 7 (since 35y = 7(5y)

Since 21x and 35y are multiples of 7, the sum 21x + 35y must be a multiple of 7.
Since only 1 answer choice (D) is a multiple of 7, D must be the correct answer.

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by Max@Math Revolution » Thu Sep 06, 2018 12:29 am
=>

21x + 35y = 7*3x + 7*5y = 7(3x+5y) is a multiple of 7. The smallest positive multiple of 7 is 7. So, we look for x and y such that 21x + 35y = 7. This will occur when 3x + 5y = 1.
When x = 2 and y = -1, 3x + 5y = 3*2 + 5*(-1) = 6 - 5 = 1, and 21x + 35y = 7(3x+5y) = 7(1) = 7.
Thus, 7 is the answer.

Therefore, the answer is D.
Answer: D
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by Scott@TargetTestPrep » Fri Sep 07, 2018 4:18 pm
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If x and y are integers, what is the least possible positive value of 21x+35y?

A. 1
B. 3
C. 5
D. 7
E. 9
Since x and y are integers and 21x + 35y = 7(3x + 5y), we see that the answer must be a multiple of 7. Therefore, D is the correct answer.

Answer: D

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