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28 Aug 2022, 00:15
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Hi,
I have a doubt in the number properties how do we find out REMAINDER FOR A LARGE EXPONENT? example 92^60/15. In this example, if we have to find the remainder of the given equation how do we solve it?
Thanks
I have a doubt in the number properties how do we find out REMAINDER FOR A LARGE EXPONENT? example 92^60/15. In this example, if we have to find the remainder of the given equation how do we solve it?
Thanks
01 Sep 2022, 21:32
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samarth1222
Hey! so Remainder for large exponents, especially questions of the format similar to the one you posted need a little playing around with numbers
Ok so we have \(\frac{92^{60}}{15}\): Now, in similar questions, we need to modify the numerator into a sum of two numbers so that one of those numbers is divisible by the denominator and then we have a smaller number to deal with (92 = 90 + 2)
\(\frac{92^{60}}{15} = \frac{(90+2)^{60}}{15}\)
\(\frac{90^{60}}{15} + \frac{2^{60}}{15}\)
Now, we know that 90 is divisible by 15 so that part will leave 0 as remainder, and we are left with \(2^{60}\) divided by 15
Let us see remainders of powers of 2, when divided by 15 to see if there is a pattern or not
\(2^1 = 2\); Remainder = \(2\)
\(2^2 = 4\); Remainder = \(4\)
\(2^3 = 8\); Remainder = \(8\)
\(2^4 = 16\); Remainder = \(1\)
\(2^5 = 32\); Remainder = \(2\)
\(2^6 = 64\); Remainder = \(4\)
\(2^7 = 128\); Remainder = \(8\)
\(2^8 = 256\); Remainder = \(1\)
As we can see there is a pattern forming for remainders of power of 2, when divided by 15 : {2,4,8,1} repeating with a cyclicity of 4
So 60 = 4*15
So 2^60 divided by 15 will leave us a remainder of 1
Answer = 1
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