What is the remainder?

Number Theory Level 1

What is the least positive integer which should be added to 2497 to make it exactly divisible by 13?

9 11 12 1

2 solutions

Sravanth C.
Jun 27, 2015

$2497 \equiv 1 \pmod{13} \\ 2497-1\equiv 0 \pmod{13} \\ 2496\equiv 0 \pmod{13}$

So, we see that $2496$ is divisible by $13$ . Hence the next number that will be divisible by $13$ is $2496+13=2509$ .

Hence the least number to be added to make $2497$ divisible by $13$ is $2509-2497=\boxed{12}$

Moderator note:

Simple standard approach.

You can improve your argument by explicitly getting the modular result for the integer to be added. Consider that integer as $x$ . The problem states,

$\begin{aligned}2497+x\equiv 0\pmod{13}&\iff 1+x\equiv 0\pmod{13}\\&\iff x\equiv -1\equiv 12\pmod{13}\end{aligned}$

Hence, the solution set for $x$ is given by $\{x:x=13k+12~\forall~k\in\Bbb Z\}$ . The problem asks for the smallest positive $x$ which is obviously attained at $k=0$ which gives the answer as $\boxed{12}$ .

Prasun Biswas - 5 years, 11 months ago

Thanks sir! That was a very good approach. $\huge\ddot\smile$

Sravanth C. - 5 years, 11 months ago

Azadali Jivani
Sep 1, 2015

2497/13 =192.0769231
0.0769231 * 7 =1
13 - 1 = 12
2497 + 12 = 2509/13 = 193
2509 is exactly divisible by 13
12(Ans.)

What is the remainder?

2 solutions

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