Question

Find the largest number which divides 70 and 125 leaving remainder 5 and 8 respectively.

Answer 1

Hint:
We know that \[{\text{dividend = divisor}} \times {\text{quotient + remainder}}\] to find the two numbers that are completely divisible by the given divisors. Then, find the HCF of these two numbers for the required answer.

Complete step by step solution:
When 70 is divided by a number say $x$, it leaves the remainder 5.
That is,
$
  {\text{70 = }}x \times {\text{quotient + 5}} \\
   \Rightarrow 70 - 5 = x \times {\text{quotient}} \\
   \Rightarrow 65 = x \times {\text{quotient}} \\
$
65 is completely divisible by the required number.
Similarly, $125 - 8 = 117$ is completely divisible by the required number.
Now, the required number is the HCF of 65 and 117.
Write the number 65 as the product of its primes.

$65 = 5 \times 13$
Similarly, write the number 117 as the product of its primes.

$117 = 3 \times 3 \times 13$
Therefore, the HCF is 13

Thus, the largest number which divides 70 and 125 leaving remainder 5 and 8 respectively is 13.

Note:
We can also calculate the HCF by long division method or factor tree method. HCF of numbers gives the largest number that divides all the given numbers.