Compute 515 modulo 7 and 713 modulo 11.
The Correct Answer and Explanation is:
To compute the given moduli, we need to apply the modulus operation. The modulus operation finds the remainder when one number is divided by another.
1. 515 modulo 7
To calculate 515mod 7515 \mod 7515mod7, we divide 515 by 7 and find the remainder.
Step 1: Divide 515 by 7:515÷7=73 remainder 4515 \div 7 = 73 \text{ remainder } 4515÷7=73 remainder 4
So, the remainder when 515 is divided by 7 is 4. Thus:515mod 7=4515 \mod 7 = 4515mod7=4
2. 713 modulo 11
Now, let’s calculate 713mod 11713 \mod 11713mod11. We divide 713 by 11 and find the remainder.
Step 1: Divide 713 by 11:713÷11=64 remainder 9713 \div 11 = 64 \text{ remainder } 9713÷11=64 remainder 9
So, the remainder when 713 is divided by 11 is 9. Thus:713mod 11=9713 \mod 11 = 9713mod11=9
Explanation:
The modulus operation is essentially the process of determining how much is “left over” after division. When we say amod ba \mod bamodb, we are looking for the remainder when aaa is divided by bbb.
For instance:
- When dividing 515 by 7, we get a quotient of 73 and a remainder of 4. Hence, 515mod 7=4515 \mod 7 = 4515mod7=4.
- Similarly, when dividing 713 by 11, we get a quotient of 64 and a remainder of 9. Hence, 713mod 11=9713 \mod 11 = 9713mod11=9.
These computations are crucial in fields like number theory, cryptography, and computer science, where the modulus operation is frequently used.
