UPSC CDS II 2024 Math Paper

Concept:The remainder of a number divided by $5$ is simply its unit digit, because $5$ divides multiples of $10$.Thus, we need the unit digit of the sum $111^{222} + 222^{333} + 333^{444}$.Explanation:Step 1: Find the unit digit of $111^{222}$.The unit digit of $111$ is $1$. Any power of $1$ ends in $1$. So the unit digit is $1$.Step 2: Find the unit digit of $222^{333}$.The unit digit of $222$ is $2$. The pattern for powers of $2$ repeats every $4$: $2^1=2$, $2^2=4$, $2^3=8$, $2^4=6$, then $2^5=2$, etc.Divide exponent $333$ by $4$: $333 \div 4$ gives remainder $1$. So the unit digit is the first in the cycle, which is $2$.Step 3: Find the unit digit of $333^{444}$.The unit digit of $333$ is $3$. The pattern for powers of $3$ repeats every $4$: $3^1=3$, $3^2=9$, $3^3=7$, $3^4=1$, then $3^5=3$, etc.Divide exponent $444$ by $4$: $444 \div 4$ gives remainder $0$ (meaning the last digit in the cycle). So the unit digit is $1$.Step 4: Add the unit digits: $1 + 2 + 1 = 4$.The sum ends with $4$, so when divided by $5$, the remainder is $4$.Answer:$4$ (Option D)