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Theory: To find the remainder of a number by 10 we just need to find the units digit of the number. And the units digit is the remainderExample:
59/5 remainder is 4 (9/5 remainder is 4)
86/10 remainder is 6 (6/10 remainder is 6)
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To find the remainder of \(3^{(4x + 2)} + 7\) by 10 we need to find the units digit of \(3^{(4x + 2)} + 7\)Let's find out the units digit of \(3^{(4x + 2)}\) first
To find the units digit of power of 3 we need to check the cyclicity in the units digit of powers of 3
\(3^1\) units’ digit is 3 | \(3^5\) units’ digit is 3
\(3^2\) units’ digit is 9 | \(3^6\) units’ digit is 9
\(3^3\) units’ digit is 7 | \(3^7\) units’ digit is 7
\(3^4\) units’ digit is 1 | \(3^8\) units’ digit is 1
Looking at the above pattern it is clear that units digit of 3 repeats after every 4th power. So, cyclicity of units digit of power of 3 is 4
=> To find units digit of \(3^{(4x + 2)}\) we need to divide the power of 3 by 4 and check what is the remainder
Now, Remainder of 4x + 2 when divided by 4 will be 2 as 4x is divisible by 4
=> Units digit of \(3^{(4x + 2)}\) = Units digit of 3^{Remainder Of 4x+2 Divided By 4} = Units digit of \(3^2\) = 9
=> Remainder of \(3^{(4x + 2)} + 7\) by 10 = Units digit of \(3^{(4x + 2)} + 7\) = Units digit of (9 + 7) = 6
So,
Answer will be DHope it helps!
Watch the following video to learn the Basics of Remainders _________________