Find the largest number which gives out remainders 2 and 5 respectively when…

To find the largest number that divides 80 and 122 with specific remainders, you must first adjust the original numbers so they are perfectly divisible by the unknown number, and then find their Highest Common Factor (H.C.F.).

Step-by-Step Calculation
1. Adjust the numbers:
Subtract the given remainders from the original numbers:

80 - 2 = 78

122 - 5 = 117

2. Find the H.C.F. of the adjusted numbers:
We need the H.C.F. of 78 and 117.

Prime factorization of 78: 2 * 3 * 13

Prime factorization of 117: 3 * 3 * 13 = 3^2 * 13

The common factors are 3 and 13.

H.C.F. = 3 * 13 = 39.