Question

Prove that no integers x,y exist satisfying x+y=100\ and\ \left(x,y\right)=3.

Answer 1

Answer:

Step-by-step explanation:

GCD of x,y = 3

x = 3k, y = 3m

3k+3m = 100

3(m+k) = 100

100 is not a multiple of 3, so it is not possible that 3(m+n) = 100

Answer 2

Given : pairs of integers x, y satisfying x+y=100 and gcd (x, y)=3

To find : prove that there are no such pairs of integers

Solution:

gcd = greatest common divisor or HCF - Highest common factor.

gcd (x, y)=3

=> x = 3A  and  y = 3B

where A and B are integers and co prime  

x + y = 100

=> 3A + 3B = 100

=> 3(A + B) = 100

=> A + B = 100/3

LHS is integer  while RHS is not integer

Hence it is not possible

So Proved that there are no pairs of integers x, y satisfying x+y=100 and gcd (x, y)=3

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