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# What is the remainder when a positive integer t is divided by 15?

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Joined: 25 Dec 2018
Posts: 146
Location: India
GMAT 1: 490 Q47 V13
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What is the remainder when a positive integer t is divided by 15?  [#permalink]

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06 Jan 2019, 04:25
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What is the remainder when a positive integer t is divided by 15?

(1) When t is divided by 45, the remainder is 13.

(2) When t is divided by 10, the remainder is 3 and when t is divided by 6, the remainder is 1.
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What is the remainder when a positive integer t is divided by 15?  [#permalink]

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Updated on: 06 Jan 2019, 07:09
What is the remainder when a positive integer t is divided by 15?

(1) When t is divided by 45, the remainder is 13.
t%45 = 13
But, 15*3 is 45, so all the numbers divisible by 45 are divisible by 15
After dividing by 45 reminder is 13 < 15, so even after dividing by 15 reminder would be 13
As reminder < quotient
Sufficient

(2) When t is divided by 10, the remainder is 3 and when t is divided by 6, the remainder is 1.
t%10 = 3
So, t can take values 13,23,33,43,53,63,73,.......
t%6 = 1
So, t can take values 7,13,19,25,...43....73.....

Common among two are 43,73.....
43%15 = 13
73%15 = 13
..........

Sufficient

Option D is correct

Originally posted by Dare Devil on 06 Jan 2019, 07:08.
Last edited by Dare Devil on 06 Jan 2019, 07:09, edited 1 time in total.
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What is the remainder when a positive integer t is divided by 15?  [#permalink]

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06 Jan 2019, 07:09
1
akurathi12 wrote:
What is the remainder when a positive integer t is divided by 15?

(1) When t is divided by 45, the remainder is 13.

(2) When t is divided by 10, the remainder is 3 and when t is divided by 6, the remainder is 1.

Very beautiful problem, akurathi12 ... Congrats and kudos!

$$t \ge 1\,\,{\mathop{\rm int}} \,\,\left( * \right)$$

$$t = 15M + R$$

$$M\mathop \ge \limits^{\left( * \right)} 0\,\,{\mathop{\rm int}} \,\,,\,\,\,0 \le R\,\,{\mathop{\rm int}} \,\, \le 14$$

$$? = R$$

$$\left( 1 \right)\,\,t = 45K + 13 = 15\left( {3K} \right) + 13\,\,,\,\,K\mathop \ge \limits^{\left( * \right)} 0\,\,{\mathop{\rm int}} \,\,\,\, \Rightarrow \,\,\,\,\left\{ \matrix{ \,M = 3K \hfill \cr \,R = 13\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}. \hfill \cr} \right.$$

$$\left( 2 \right)\,\,\left\{ \matrix{ \,t = 10J + 3\,\,\,,\,\,\,J\mathop \ge \limits^{\left( * \right)} 0\,\,{\mathop{\rm int}} \,\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,\,3} \,\,\,\,\,\,3t = 30J + 9 \hfill \cr t = 6L + 1\,\,\,,\,\,\,L\mathop \ge \limits^{\left( * \right)} 0\,\,{\mathop{\rm int}} \,\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,5} \,\,\,\,\,\,5t = 30L + 5 \hfill \cr} \right.\,\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( - \right)} \,\,\,\,\,2t = 30\left( {L - J} \right) - 4$$

$$2t = 30\left( {L - J} \right) - 4\,\,\,\,\, \Rightarrow \,\,\,\,\,t = 15\left( {L - J} \right) - 2 = 15\left( {L - J - 1} \right) + 15 - 2\,\,\,\,\, \Rightarrow \,\,\,\,\left\{ \matrix{ \,M = L - J - 1 \hfill \cr \,R = 13\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}. \hfill \cr} \right.$$

The correct answer is therefore (D).

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

Regards,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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Re: What is the remainder when a positive integer t is divided by 15?  [#permalink]

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06 Jan 2019, 07:20
fskilnik wrote:
akurathi12 wrote:
What is the remainder when a positive integer t is divided by 15?

(1) When t is divided by 45, the remainder is 13.

(2) When t is divided by 10, the remainder is 3 and when t is divided by 6, the remainder is 1.

Very beautiful problem, akurathi12 ... Congrats and kudos!

$$t \ge 1\,\,{\mathop{\rm int}} \,\,\left( * \right)$$

$$t = 15M + R$$

$$M\mathop \ge \limits^{\left( * \right)} 0\,\,{\mathop{\rm int}} \,\,,\,\,\,0 \le R\,\,{\mathop{\rm int}} \,\, \le 14$$

$$? = R$$

$$\left( 1 \right)\,\,t = 45K + 13 = 15\left( {3K} \right) + 13\,\,,\,\,K\mathop \ge \limits^{\left( * \right)} 0\,\,{\mathop{\rm int}} \,\,\,\, \Rightarrow \,\,\,\,\left\{ \matrix{ \,M = 3K \hfill \cr \,R = 13\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}. \hfill \cr} \right.$$

$$\left( 2 \right)\,\,\left\{ \matrix{ \,t = 10J + 3\,\,\,,\,\,\,J\mathop \ge \limits^{\left( * \right)} 0\,\,{\mathop{\rm int}} \,\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,\,3} \,\,\,\,\,\,3t = 30J + 9 \hfill \cr t = 6L + 1\,\,\,,\,\,\,L\mathop \ge \limits^{\left( * \right)} 0\,\,{\mathop{\rm int}} \,\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,5} \,\,\,\,\,\,5t = 30L + 5 \hfill \cr} \right.\,\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( - \right)} \,\,\,\,\,2t = 30\left( {L - J} \right) - 4$$

$$2t = 30\left( {L - J} \right) - 4\,\,\,\,\, \Rightarrow \,\,\,\,\,t = 15\left( {L - J} \right) - 2 = 15\left( {L - J - 1} \right) + 15 - 2\,\,\,\,\, \Rightarrow \,\,\,\,\left\{ \matrix{ \,M = L - J - 1 \hfill \cr \,R = 13\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}. \hfill \cr} \right.$$

The correct answer is therefore (D).

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

Regards,
Fabio.

Thank you so much for the explanation.. especially for Statment 1, I am unable to understand.. not its clear
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Re: What is the remainder when a positive integer t is divided by 15?  [#permalink]

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06 Jan 2019, 07:23
akurathi12 wrote:
Thank you so much for the explanation.. especially for Statment 1, I am unable to understand.. not its clear

My pleasure, akurathi12 !

Thank you for the kudos and see you in other posts.

Regards and success in your studies!
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
Re: What is the remainder when a positive integer t is divided by 15?   [#permalink] 06 Jan 2019, 07:23
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