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

Source
Suggested Reading

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 \(1+1=2\)

Subscribe to Cheenta at Youtube