The sum of two numbers is 1212 and their HCF is 101; find the numbers?

To find the two numbers given that their sum is 1212 and their Highest Common Factor (HCF) is 101, we'll use some basic number properties of HCF and sums.
Let's denote the two numbers as a and b , such that:

a + b = 1212
HCF ( a , b ) = 101

Given the HCF is 101, this means that both a and b are multiples of 101. Therefore, we can express:
a = 101 x b = 101 y
where x and y are integers.
Now replace a and b in the sum equation:
a + b = 101 x + 101 y = 1212
We can factor out 101 from the equation:
101 ( x + y ) = 1212
To find x + y , divide both sides by 101:
x + y = 101 1212 ​
Simplifying gives:
x + y = 12
Now, we need to find two numbers x and y that add up to 12. Let's assume the possibilities are such that x = 5 and y = 7 (because 5 and 7 are manageable integer pairs that sum up to 12).
Using these values:

When x = 5 , a = 101 × 5 = 505
When y = 7 , b = 101 × 7 = 707

Checking:

a + b = 505 + 707 = 1212

This confirms that the two numbers are 505 and 707 . Therefore, the numbers are 505 and 707 .

Answered by IsabellaRoseDavis | 2025-07-21