Question

X and Y are the digits at the unit's place of the numbers (408X) and (789Y) where X ≠ Y. However, the digits at the unit's place of the numbers $(408X)^{63}$ and $(789Y)^{85}$ are the same. What will be the possible value(s) of (X + Y)?

Accepted Answer

B. Exponents modulo 4: $63 \equiv 3 \pmod 4$, $85 \equiv 1 \pmod 4$. We need unit digit of $X^3 = Y^1 = Y$. Checking pairs $(X, Y)$ where $X 
eq Y$ and $X^3 \equiv Y \pmod{10}$: - $X = 2 \implies Y = 8$ (sum = 10) - $X = 3 \implies Y = 7$ (sum = 10) - $X = 7 \implies Y = 3$ (sum = 10) - $X = 8 \implies Y = 2$ (sum = 10) In all possible valid cases, $X + Y = 10$.

Cyclicity of Unit Digits

Explanation

Practise this under exam conditions