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

Question

(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.

Dare Devil · Answer

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

fskilnik · Answer

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.

akurathi12 · Answer

Thank you so much for the explanation.. especially for Statment 1, I am unable to understand.. not its clear

fskilnik · Answer

My pleasure,  akurathi12  !

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

Regards and success in your studies!
Fabio.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia