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,
$$\begin{gathered} \text{70 = }x\times \text{quotient + 5} \\ \Rightarrow 70-5=x\times \text{quotient} \\ \Rightarrow 65=x\times \text{quotient} \end{gathered}$$
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.