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Find Remainder when
$(13 {}^{13} + 1)$
is divided by 14

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Answer 1

▪Answer :- 0

▪Explanation :-

《 Refer To Attachment 》

Answer 2

Given :- Find the remainder when (13¹³ + 1) is divided by 14 ?

Solution :-

→ (13¹³ + 1) ÷ 14

→ (13¹³ + 1¹³) ÷ 14 { since 1^n = 1 always .}

→ (13¹³ + 1¹³) ÷ (13 + 1)

→ (a^n + b^n) ÷ (a + b)

when ,

• n = odd => Remainder will be 0 .
• Or we can say that (a^n + b^n) is completely divisible by (a + b) when n is odd .

since n = 13 = odd number ,

therefore,

→ (13¹³ + 1¹³) ÷ (13 + 1) = Remainder 0 (Ans.)

Extra :-

• (a^n - b^n) ÷ (a + b) = Remainder 0 , when n is even .
• (a^n - b^n) ÷ (a - b) = Remainder 0 , for all values of n .

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