Latest Maths Olympiad Aptitude Question SOLUTION: what is the remainder when 6^17+17^6 is divided by 7

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(#M40039355) MATHS OLYMPIAD Category question POWER Keep an EYE

what is the remainder when
6^17+17^6 is divided by 7

Asked In Maths Olympiad NAGARAJAN (9 years ago)

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Farzand Bawa  (9 years ago)

(6^17+17^6)/7 can be written as (-1)^17/7 +(3)^6/7 using remainder theorem.
Therefore, it can be seen as -1 (as odd power of -1 yields -1) + ((9)^3/7) again using remainder theorem.
Now again applying remainder theorem can see it gives us -1+((2)^3/7)=-1+(8/7)=-1+1=0. Hence the remainder is 0 when 6^17+17^6 is divided by 7.
Hopefully this was helpful.

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Sudheesh kuamr  (9 years ago)

As we know (an + 1)^x / a --> 1 rem or (an - 1)^ x / a --> -1 rem
for example: 27^3/7 -->(28 - 1)^3 / 7 --> -1 rem
8^3/7 -->(7 + 1)^3 / 7 --> 1 rem
Now to question:
6^17 / 7 --> (7 - 1)^17 / 7 --> (-1)^17 / 7 --> -1 rem
17^6 / 7 --> (3)^6 / 7 --> (27)^4 / 7 --> (28 - 1)^4 / 7 --> (-1)^4 / 7 --> 1 rem (even power)
So, -1 + 1 --> 0 ans

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