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If a and b are positive integers such that when a is divided

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by Max@Math Revolution » Thu Oct 18, 2018 12:03 am

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[Math Revolution GMAT math practice question]

If a and b are positive integers such that when a is divided by b, the remainder is 10, what is the value of b?

1) b > 10
2) b < 12
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Source: — Data Sufficiency |
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by fskilnik@GMATH » Thu Oct 18, 2018 5:32 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If a and b are positive integers such that when a is divided by b, the remainder is 10, what is the value of b?

1) b > 10
2) b < 12
Nice conceptual problem, Max. Congrats!
$$a,b\,\, \ge 1\,\,\,{\rm{ints}}$$
$$a = Qb + 10\,\,\,\,\,\left\{ \matrix{
\,Q\,\,\,{\mathop{\rm int}} \hfill \cr
\,{\rm{remainder}}\,\,10\,\,\, \Rightarrow \,\,\,\,b \ge 11\,\, \hfill \cr} \right.$$
$$? = b$$
$$\left( 1 \right)\,\,{\rm{already}}\,\,{\rm{known}}\,\,\, \Rightarrow \,\,\,{\rm{INSUFF}}.$$
$$\left( 2 \right)\,\,\left\{ \matrix{
\,b < 12 \hfill \cr
\,b \ge 11 \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,b = 11\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.$$

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

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Max@Math Revolution » Sun Oct 21, 2018 5:40 pm
=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

By the quotient-remainder theorem, we can write a = b * q + 10, where the remainder 10 is less than b, that is, b > 10.

Thus, condition 2) "b<12" is sufficient since it gives the unique solution b = 11.

Note: Condition 1) does not give a unique solution. For example, we might have b = 11 or b = 12. Thus, it is not sufficient.

Therefore, B is the answer.
Answer: B
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