Which numbers will divide into both 90 and 100 with no remainder?

x is the divisor

y is the result

r is the remainder  

if the remainder is always 10, then the number of 100 - 10 = 90 must be divisible without a remainder.  

our equation becomes:  

90/x = y with no remainder.  

multiply both sides of this equation by x to get:  

90 = x*y  

find all the factors of x and you will be able to find the anser.  

90 equals:

1*90

2*45

3*30

6*15

9*10  

now take these factors and divide them into 100.  

100/1 = 100 with no remainder

100/90 = 1 with a remainder of 10

100/2 = 50 with no remainder

100/45 = 2 with a remainder of 10

100/3 = 33 with a remainder of 1

100/30 = 3 with a remainder of 10

100/6 = 16 with a remainder of 4

100/15 = 6 with a remainder of 10

100/9 = 11 with a remainder of 1

100/10 = 10 with a remainder of 0.  

narrow it down to all those with a remainder of 10 and you get:  

100/90

100/45

100/30

100/15  

the divisor had to be greater than 10.  

the smallest divisor is 15.  

the largest divisor is 90.  

your answer is:  

the sum of the smallest and the largest divisor is 90 + 15 = 105.