Remainder of 13*17 when divided by 12

To find the remainder of 13*17 when divided by 12, we can use the concept of modular arithmetic. Modular arithmetic deals with remainders when dividing one number by another.

Method 1: Direct Calculation
1. Multiply 13 and 17: 13 * 17 = 221.
2. Divide the product by 12: 221 ÷ 12 = 18 remainder 5.
3. Therefore, the remainder of 13*17 when divided by 12 is 5.

Method 2: Using Modular Arithmetic
1. Express 13*17 as a congruence in terms of modular arithmetic: 13*17 ≡ x (mod 12), where x is the remainder we want to find.
2. Break down the product into smaller congruences: 13 ≡ 1 (mod 12) and 17 ≡ 5 (mod 12).
3. Substitute the congruences into the original equation: 1 * 5 ≡ x (mod 12).
4. Simplify the equation: 5 ≡ x (mod 12).
5. Therefore, the remainder of 13*17 when divided by 12 is 5.

Method 3: Divisibility Rule of 12
1. The divisibility rule of 12 states that if a number is divisible by both 3 and 4, it is divisible by 12.
2. Check if the product 13*17 is divisible by 3: 13 + 17 = 30, which is divisible by 3.
3. Check if the product 13*17 is divisible by 4: 221 is not divisible by 4.
4. Since 221 is not divisible by 12, find the remainder by subtracting the largest multiple of 12 less than 221: 221 - 12 = 209.
5. Therefore, the remainder of 13*17 when divided by 12 is 209.

Conclusion:
The remainder of 13*17 when divided by 12 can be found using various methods. In this case, the remainder is 5.