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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

But try the problem first…

Source

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 $1+1=2$