Try this beautiful problem from Integer based on Remainder useful for ISI B.Stat Entrance.

## Remainder Problem | ISI B.Stat Entrance | Problem-90

The remainder when \( 3^{12} +5^{12}\) is divided by 13 is……

- 2
- 1
- 4
- 3

**Key Concepts**

Division algorithm

Divisor

Number theory

## Check the Answer

But try the problem first…

Answer: 2

TOMATO, Problem 90

Challenges and Thrills in Pre College Mathematics

## Try with Hints

First hint

The given number is \( 3^{12} +5^{12}\)

we have to check if it is divided by 13 what will be the remainder? if we express the number in division algorithm form then we have……..\( 3^{12} +5^{12}=((3)^3)^4+((5)^2)^6)=(27)^4 +(25)^6\)=\(((13 \times 2+1)^4+(13 \times 2-1)^6)\)

Can you now finish the problem ……….

Second Hint

**Remainder :**

Clearly if we divide \(((13 \times 2+1)^4+(13 \times 2-1)^6)\) by 13 then from \((13 \times 2+1)^4\) , the remainder be 1 and from \((13 \times 2-1)^6)\), the remainder is 1

can you finish the problem……..

Final Step

Therefore the total remainder is \(1+1=2\)

## Other useful links

- https://www.cheenta.com/area-of-the-figure-amc-8-2014-problem-20/
- https://www.youtube.com/watch?v=V01neV8qmh4

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