Question

The largest number which divides 70 and 125, leaving 5 and 8, respectively, is
a) 13
b) 65
c) 875
d) 1750

Answer 1

Hint: We have two numbers 70 and 125 which are to be divided by the same number leaving remainders 5 and 8, respectively. So, the numbers to be divided are:
$\begin{array}{rl}& 70-5=65\\ & 125-8=117\end{array}$
We need to find a common number that divides 65 as well as 117 leaving no remainder.
So, we need to find HCF of 65 and 117.

Complete step by step answer:
Since it is given that we have to find the largest number that divides 70 and 125 each, it means that we need to find the highest common factor, i.e. HCF. That leaves 5 and 8 as the remainder respectively.
By prime-factorization method;
The prime factors of 65 are: $$65=5\times 13$$
And the prime factors of 117 are: $$117=3\times 3\times 13$$
Hence, the common prime factor of 65 and 117 is 13.

Now, dividing 70 by 13 we get 5 as remainder
$$13\stackrel{5}{\stackrel{―}{)\begin{array}{rl}& 70\\ & 65\\ & \stackrel{―}{05}\end{array}}}$$
Similarly, dividing 125 by 13 we get 8 as remainder
$$13\stackrel{9}{\stackrel{―}{)\begin{array}{rl}& 125\\ & 117\\ & \stackrel{―}{008}\end{array}}}$$
Therefore, 13 is the largest number that divides 70 and 125 and leaves 5 and 8 as the remainder respectively.

So, the correct answer is “Option A”.

Note: While finding the largest number that divides two numbers, some might confuse whether to calculate HCF or LCM. So, always go for HCF when finding the largest number.
Also, while factoring, take care of using prime-numbers only. After getting the common factors, recheck by dividing the numbers with the factor to be assured that the answer is correct.