Question

find all possible pair of two natural number whose sum is77 and their gcd is 7

Answer 1

Given that the GCD of two natural numbers is 7 and sum = 77.

Possible pair of natural numbers are:

(1) 7 and 70.

Sum = 7 + 70 = 77.

Prime factorization of 7 = 7

Prime factorization of 70 = 2 * 5 * 7

GCD(7,70) = 7.



(2) 14 and 63

Sum = 14 + 63 = 77.

Prime factorization of 14 = 2 * 7

Prime factorization of 63 = 3 * 3 * 7

GCD(14,63) = 7.



(3) 21 and 56

Sum = 21 + 56 = 77

prime factorization of 21 = 3 * 7

Prime factorization of 56 = 2 * 2 * 2 * 7

GCD(21,56) = 7.



(4) 28 and 49.

Sum = 28 + 49

Prime factorization of 28 = 2 * 2 * 7

Prime factorization of 49 = 7 * 7

GCD(28,49) = 7


(5) 35 and 42.

Sum = 35 + 42 = 77

Prime factorization of 35 = 5 * 7

Prime factorization of 42 = 2 * 3 * 7

GCD(35,42) = 7.


There are 5 such possible pairs.



Hope this helps!

Answer 2

Given,
The G.C.D of 2 natural nos. is 7 & sum is 77.

Possible pairs of natural nos. are : -

i ) First pair possible = 14 & 63
Sum = 14 + 63 = 77

The only possible prime factor = 7 in 14 & 63.
Therefore H.C.F = 7

i i ) Second pair possible = 21 & 56
Sum = 21 + 56 = 77

The only possible prime factor = 7 in 21 & 56.
Therefore H.C.F = 7

i i i ) Third possible pair = 28 & 49
Sum = 28 + 49 = 77

The only possible prime factor = 7 in 28 & 49
Therefore H.C.F = 7

i v ) Fourth pair possible = 35 & 42
Sum = 35 + 42 = 77

The only possible prime factor = 7 in 35 & 42
Therefore H.C.F = 7

v ) Fifth possible pair = 7 + 70 = 77
Sum = 70 + 7= 77

The only possible prime factor = 7 in 70 & 7.
Therefore H.C.F = 7

Therefore there are total 5 pairs possible.

Hope this helps !