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Find the remainder when 2^(100)+3^(100)+...

Find the remainder when `2^(100)+3^(100)+4^(100)+5^(100)` is divided by 7.

Text Solution

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`(2^100+3^100+4^100+5^100)/7`
`=(2*2^99+(9^(1/2))^100+2^200+5*5^99)/7`
`=(2*(2^3)^33+9^50+2^2*(2^3)^66+5*(5^3)^33)/7`
`=(2*8^33+9^50+2^2*8^66+5*(125)^33)/7`
`=(2*(1)^33+(2)^50+2^2*(1)^66+5*(-1)^33)/7`
`=(2*(1)^33+2^2*(8)^16+2^2*(1)^66+5*(-1)^33)/7`
`=(2*(1)^33+2^2*(1)^16+2^2*(1)^66+5*(-1)^33)/7`
`=(2+4+4-5)/7 = 5/7`
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