Question

The biggest number which divides 125 and 70, and gives remainder 8 and 5 respectively will be:

• A. 15
• B. 13
• C. 17
• D. 14

Hint:

When solving problems involving remainders, subtract the remainders from the numbers and find the GCD of the results.

Answer 1

The Correct Option is D

Step 1: Understand the problem.
We need to find the largest number \( x \) that divides both 125 and 70, and leaves remainders of 8 and 5, respectively. This means: \[ 125 \equiv 8 \pmod{x} \quad \text{and} \quad 70 \equiv 5 \pmod{x} \]
Step 2: Simplify the equations.
This implies that: \[ 125 - 8 = 117 \quad \text{and} \quad 70 - 5 = 65 \] Thus, \( x \) must divide both 117 and 65. We now find the greatest common divisor (GCD) of 117 and 65.
Step 3: Find the GCD.
The prime factorizations of 117 and 65 are: \[ 117 = 3 \times 3 \times 13 \] \[ 65 = 5 \times 13 \] The common factor is 13, so the GCD of 117 and 65 is 13.
Step 4: Conclusion.
Therefore, the biggest number that divides both 125 and 70 and gives the required remainders is \( 13 \), and the correct answer is (D) 14.