A 4-digit number is formed by repeating a 2-digit number such as 2525, 3232 etc. Any number of this form is exactly divisible by

## Concept & Logic When a multi-digit number is formed by repeating a smaller block of digits, it can be factored using place value expansion. ## Step-by-Step Solution * Let the 2-digit number being repeated be represented algebraically as $xy$. * Its base-10 value is $(10x + y)$. * The 4-digit number formed by repeating it is $xyxy$. * Expand this 4-digit number using place values: $$ 1000x + 100y + 10x + y $$ * Group the $x$ terms and $y$ terms together: $$ 1010x + 101y $$ * Factor out the common term, 101: $$ 101(10x + y) $$ * This shows the number is always a multiple of 101, regardless of what $x$ and $y$ are. * 101 is the smallest 3-digit prime number. ## Exam Strategy & Shortcut To quickly find the repeating multiplier, divide the repeated number by the base repeating block. Take 2525 and divide it by 25. $$ 2525 \div 25 = 101 $$ 101 has exactly three digits and is prime. ## Common Pitfall A common mistake is assuming the number is divisible by 11 because 11 is a known divisor for numbers like 1221 (palindromes) or numbers with alternating digit sums. However, $2525 \div 11 = 229.54$, which is not an integer. ## Final Answer **Therefore, the correct answer is Smallest 3-digit prime number.**