Question

Find the sum of all 2 digit numbers N=ab¯¯¯, where a≠0, such that N divides a0b¯¯¯¯¯.

Clarification: The notation a0b¯¯¯¯¯ used in the question means that we read it as a 3-digit number. For example, with a=4,b=8, a08¯¯¯¯¯ represents the number 408, and not the number 4×0×8=0.

Answer 1

(100a+b)/(10a+b)=n where 'n' is integer.

Let, b=0 so, 100a/10a=10 here 'a' can take 9 values from 1 to 9 so, we get 10+20+30.....+90=450
'a' not equal to 'b' cause 101a/11a is not an integer.

To find more values, we have to do some calculations.

100a+b=10na+nb
=>10a(10-n)=b(n-1)
=>10a/b=(n-1)/(10-1)

when n=6
10a/b=5/4
=>40a=5b
=>8a=b
=>a/b=1/8

so, a=1 , b=8 (a,b can not take value greater than 9). Get the number 18
in this way, trying with n=7, 9 get the numbers 15, 45 (i tried the values 1 to 9 and got for only n=6,7,9)

so the sum is 450+15+18+45=528