Q. 115.0( 2 Votes )

Find all digits x, y such that is divisible by 36.

Answer :

The number divisible by 36 is also divisible by 4 & 9.

Since, 36 = 4 × 9


For divisibility by 4,


The last two digits of the number should be divisible by 4 for the number to be divisible by 4.


The last two digits of the number 34x5y is 5y.


5y should be divisible by 4.


y = 2 or y = 6


(0 ≤ y ≤ 9)


Because 52 is divisible by 4 and 56 is also divisible by 4.


For divisibility by 9,


The sum of digits of the number should be divisible by 9 for the number to be divisible by 9.


Sum of digits of 34x5y is,


Sum = 3 + 4 + x + 5 + y


Sum = 12 + x + y


If y = 2,


Sum = 12 + x + 2


Sum = 14 + x


Now if x = 4, (0 ≤ x ≤ 9)


Then sum = 18, which is divisible by 9.


And if x = 13,


Then sum = 27, which is divisible by 9 but x ≠ 13 as 13 is a 2-digit number.


So, we have found one pair of solution. That is, (x, y) = (4, 2).


If y = 6,


Sum = 12 + x + 6


Sum = 18 + x


Now if x = 0, (0 ≤ x ≤ 9)


Then sum = 18, which is divisible by 9.


Also, if x = 9, (0 ≤ x ≤ 9)


Then sum = 27, which is divisible by 9.


So, we have found two pair of solutions. That is, (x, y) = (0, 6) and (x, y) = (9, 6).


Thus, we have x = 4, y = 2 or x = 0, y = 6 or x = 9, y = 6.


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