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If a certain positive integer is divided by 9, the remainder

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by BTGmoderatorLU » Fri Dec 28, 2018 12:45 pm

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Source: GMAT Prep

If a certain positive integer is divided by 9, the remainder is 3. What is the remainder when the integer is divided by 5?

1) If the integer is divided by 45, the remainder is 30.
2) The integer is divisible by 2.

The OA is A
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by Jay@ManhattanReview » Sat Dec 29, 2018 1:41 am
BTGmoderatorLU wrote:Source: GMAT Prep

If a certain positive integer is divided by 9, the remainder is 3. What is the remainder when the integer is divided by 5?

1) If the integer is divided by 45, the remainder is 30.
2) The integer is divisible by 2.

The OA is A
Given that the positive integer divided by 9 leaves the remainder 3, we have the number = 9q + 3; where q = quotient.

We have to determine the remainder when the integer 9q + 3 is divided by 5.

Let's take each statement one by one.

1) If the integer is divided by 45, the remainder is 30.

Again, say the number is 45p + 30; where p = quotient.

Thus, we have 9q + 3 = 45p + 30

=> q = 5p + 3

Thus, the number 9q + 3 = 9(5p + 3) + 3 = 45p + 27 + 3 = 45p + 30

The remainder when 45p + 30 divided by 5 is 0 since 45, as well as 30, are divisible by 5. Sufficient.

2) The integer is divisible by 2.

=> The integer 9q + 3 is even.

For 9q + 3 to be even, we must have q = an odd integer

Case 1: Say q = 1; thus 9q + 3 = 9*1 + 3 = 12. The remainder when 12 divided by 5 is 2.
Case 2: Say q = 3; thus 9q + 3 = 9*3 + 3 = 30. The remainder when 30 divided by 5 is 0.

No unique answer. Insufficient.

The correct answer: A

Hope this helps!

-Jay
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by fskilnik@GMATH » Sat Dec 29, 2018 5:33 pm
BTGmoderatorLU wrote:Source: GMAT Prep

If a certain positive integer is divided by 9, the remainder is 3. What is the remainder when the integer is divided by 5?

1) If the integer is divided by 45, the remainder is 30.
2) The integer is divisible by 2.
$$N \ge 1\,\,{\mathop{\rm int}} \,\,\,\left( * \right)$$
$$N = 9M + 3\,\,,\,\,\,M\mathop \ge \limits^{\left( * \right)} 0\,\,{\mathop{\rm int}} $$
$$N = 5K + R\,\,,\,\,\,K\mathop \ge \limits^{\left( * \right)} 0\,\,{\mathop{\rm int}} $$
$$0 \le R\,\,\,{\mathop{\rm int}} \le 4$$
$$? = R$$

$$\left( 1 \right)\,\,\,N = 45J + 30\,\,,\,\,\,J\,\,{\mathop{\rm int}} \,\,\,\,\, \Rightarrow \,\,\,\,N\,\,{\rm{is}}\,\,{\rm{divisible}}\,\,{\rm{by}}\,\,5\,\,\,\, \Rightarrow \,\,\,\,? = 0$$
$$\left( 2 \right)\,\,\,N\,\,{\rm{even}}\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,{\rm{M = 1}}\,\,\,\, \Rightarrow \,\,\,\,N = 12\,\,\,\, \Rightarrow \,\,\,\,? = 2 \hfill \cr
\,{\rm{Take}}\,\,{\rm{M = 3}}\,\,\,\, \Rightarrow \,\,\,\,N = 30\,\,\,\, \Rightarrow \,\,\,\,? = 0 \hfill \cr} \right.$$

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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