Question

digits a and b are such that the product 4al x 25 b is divisible by 36 (in base 10) the number of ordered pairs

Answer 1

Answer:

For the product (2A5)(13B) to be divisible by 36, we need it to be divisible by both 4 and 9. Since 2A5 is odd, it does not contain a factor of 2.Therefore, 13B must be divisible by 4.

For a positive integer to be divisible by 4, the number formed by its last two digits must

be divisible by 4, i.e. 3B is divisible by 4, i.e. B= 2 or B = 6.

Case 1: B = 2

In this case, 132 is divisible by 3, but not by 9. Therefore, for the original product to be

divisible by 9, we need 2A5 to be divisible by 3.

For a positive integer to be divisible by 3, the sum of its digits is divisible by 3, i.e.

2 + A + 5 = A + 7 is divisible by 3.

Therefore, A = 2 or 5 or 8.

Case 2: B = 6

In this case, 136 contains no factors of 3, so for the original product to be divisible by 9,

we need 2A5 to be divisible by 9.

For a positive integer to be divisible by 9, the sum of its digits is divisible by 9, i.e.

2 +A+5 is divisible by 9. Therefore, A is 2.

Therefore, the four possible ordered pairs are

⇒

A, B = 2,2, 8,2, 5,2, 2,6