Question

How many pairs of values of (x, y) exist such that the number 42xy60 is divided by 72?
(a). 2
(b). 3
(c). 4
(d). 5
(e). 6

Answer 1

- Hint: A number is divisible by 72 if and only if it is divisible by 8 and 9 because 8 and 9 are the factors of 72. A number is divisible by 9 if the sum of numbers is multiple of 9 and a number is divisible by 8 if its last three digits are divisible by 8.

Complete step-by-step solution -
First we will check divisibility of 8 in 42xy60.
We know that a number is divisible by 8 if its last three digits is divisible by 8.
We need to check that, for what values of y, y60 is divisible by 8?
 y can be any number between 0 to 9 but we need to choose only those numbers by which y60 is divisible by 8.
2 is a factor of 60 we need to find that, for what values of y, y6 is divisible by 4.
Only 16, 36, 56, 76, 96 is divisible by 4.
We can conclude that, 1, 3, 5, 7, 9 are the values of y by which y60 is divisible by 8.

Now check the divisibility of 9 in 42xy60.
We know that a number is divisible by 9 if the sum of numbers is multiple of 9.
Substitute above values of y in the number.
We will have,
42x160, 42x360, 42x560, 42x760, 42x960 here, we need to find x such that the sum of numbers is multiple of 9.
4 +2 + x+ 1+ 6 +0 =13 +x if x =5 then 42x160 is divisible by 9.
4 +2 + x+ 3+ 6 +0 =15 +x if x =3 then 42x160 is divisible by 9.
4 +2 + x+ 5+ 6 +0 =17 +x if x =1 then 42x160 is divisible by 9.
4 +2 + x+ 7+ 6 +0 =19 +x if x =8 then 42x160 is divisible by 9.
4 +2 + x+ 9+ 6 +0 =21 +x if x =6 then 42x160 is divisible by 9.

(5,1), (3,3), (1,5), (8,7), (6,9) are the values of pairs (x, y) such that the number 42xy60 is divisible by 72.
The number of pairs is 5.
Option (d) is correct.

NOTE- We can also first check divisibility for 9, then we will have a lot of possibilities that’s why we will first check divisibility for 8 then go for 9.