Question

The total number of two digit numbers 'n', such that $3^n + 7^n$ is a multiple of 10, is ________ .

Hint:

Problems involving the last digit of large powers can be solved by finding the cycle of the last digits. For sums and divisibility, modular arithmetic is a more powerful tool. The property $a+b$ divides $a^n+b^n$ for odd 'n' is useful here ($3+7=10$).

Answer 1

Correct Answer: 45

We need: \[ 3^n + 7^n \equiv 0 \pmod{10} \] Since \(7\equiv -3\pmod{10}\): \[ 3^n + (-3)^n \equiv 0 \] \[ 3^n(1+(-1)^n)\equiv 0 \] This holds only if \(n\) is odd. Two-digit odd numbers: \(11,13,\dots,99\) \[ \text{Count}=\frac{99-11}{2}+1=45 \] \[ \boxed{45} \]