Question

# The largest number, which divides 25, 73 and 97 to leave the same remainder in each case, isA) 24 B) 23 C) 21D) 6

Hint: In this question, we use the concept of HCF. The Highest Common Factor of two or more numbers is the highest number by which all the given numbers are divisible without leaving any remainders. Basically, it is the largest number that divides all the given numbers.

Given, we have three numbers 25, 73 and 97.
Since the remainders are the same the difference of every pair of given numbers would be exactly divisible by the required number.
The difference of every pair of given numbers are,
$73 - 25 = 48,97 - 73 = 24,97 - 25 = 72$
So, the required number is HCF of 24, 48 and 72.
Now, the factors of $24 = 2 \times 2 \times 2 \times 3 = {2^3} \times 3$
The factors of $48 = 2 \times 2 \times 2 \times 2 \times 3 = {2^4} \times 3$
The factors of $72 = 2 \times 2 \times 2 \times 3 \times 3 = {2^3} \times {3^2}$
Now, the Highest Common Factor (HCF) of 24, 48 and 72 is the highest number by which all the given numbers are divisible without leaving any remainders.
$HCF = {2^3} \times 3 = 8 \times 3 \\ \Rightarrow HCF = 24 \\$
The required number is 24.
So, the correct option is (a).

Note: Whenever we face such types of problems we use some important points. First we subtract the pair of given numbers then find the factors of the upcoming number. So, the HCF of the upcoming number is equal to the required number.