Try this beautiful problem from Integer based on Remainder useful for ISI B.Stat Entrance.
The remainder when \( 3^{12} +5^{12}\) is divided by 13 is......
Division algorithm
Divisor
Number theory
Answer: 2
TOMATO, Problem 90
Challenges and Thrills in Pre College Mathematics
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 ..........
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........
Therefore the total remainder is \(1+1=2\)
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