N is the smallest three digit prime number. When N is divided by 13, what will be the remainder?

Introduction / Context:
This question combines basic knowledge of prime numbers with simple division to find a remainder. It is typical of aptitude questions that test number properties and divisibility without heavy calculations.

Given Data / Assumptions:

• N is the smallest three digit prime number.
• We need the remainder when N is divided by 13.
• All calculations are in standard integer arithmetic.

Concept / Approach:
First, identify the smallest three digit prime. Then perform a simple division by 13 to find the remainder. Since 100 is composite, we test the next number 101 for primality by checking divisibility by small primes.

Step-by-Step Solution:
The smallest three digit number is 100, but 100 is not prime because 100 = 4 × 25 and also 100 = 2^2 × 5^2. The next number is 101. We must check if 101 is prime. Test divisibility by prime numbers up to the square root of 101. The square root of 101 is a little more than 10. Check divisibility by 2: 101 is odd, so not divisible by 2. Check divisibility by 3: sum of digits is 1 + 0 + 1 = 2, not a multiple of 3, so not divisible by 3. Check divisibility by 5: the last digit is 1, not 0 or 5, so not divisible by 5. Check divisibility by 7: 7 × 14 = 98 and 7 × 15 = 105, so 101 is not divisible by 7. Check divisibility by 11: 11 × 9 = 99 and 11 × 10 = 110, so 101 is not divisible by 11. Since 101 is not divisible by any prime up to 10, 101 is prime. Therefore N = 101. Now divide 101 by 13. We have 13 × 7 = 91 and 13 × 8 = 104 which is greater than 101. Thus 101 = 13 × 7 + 10, so the remainder when 101 is divided by 13 is 10.

Verification / Alternative check:
You can quickly re check by multiplying 13 by 7 to get 91 and then adding the remainder 10 to reach 101. Since this matches N exactly, the remainder is confirmed as 10.

Why Other Options Are Wrong:
7, 8, 9, 11: These are simply other possible remainders less than 13. They would correspond to incorrect quotients or mistaken multiplication. None of them satisfy 101 = 13 × integer + remainder with that remainder value.

Common Pitfalls:
One mistake is to assume that 100 might be prime or to choose 103 without properly checking 101. Another is to perform the division by 13 incorrectly and miscalculate the remainder. Carefully checking divisibility and basic multiplication prevents these errors.

Final Answer:
The remainder when N is divided by 13 is 10.