probability:

 

the total cases :

x and Y can take any value from the numbers 1 - 100 and any combinations.

so the total numbers possible is  100 * 100.

 

favourable cases:

 

the numbers x and Y will be of the form ,lets say

x = 4n+1 , 4n+2 ,4n+3,4n+4;

y= 4m+1 , 4m+2 ,4m+3,4m+4.

and note that m may be equak to n.

 

thus,

 

7^x + 7^y  may take numbers in the above form.

 

but onl numbers in the form,

 case 1 : 7⁴ⁿ⁺¹ + 7⁴ⁿ⁺³ 

and  case 2:  7 ⁴ⁿ⁺² + 7⁴ⁿ⁺⁴  will be divisible by 5.

now n and m can take values from 0 to 24.

a total of 25 numbers and in any combination.

 

thsu the total possible numbers for case 1 are 25 * 25 =625.

similarly for case 2.

 

thus a total favourable cases of 1250.

thus probability is

 

 P = 1250/ 10000.

which is exactly 1/8.