If a and b are both positive, two-digit integers, is a + b a multiple of 11? The tens digit of a is equal to the units digit of b, and the tens digit of b is equal to the units digit of a Both a and b are odd.

Question

Callum Cuevas

Mathematics

5 September, 01:32

If a and b are both positive, two-digit integers, is a + b a multiple of 11? The tens digit of a is equal to the units digit of b, and the tens digit of b is equal to the units digit of a Both a and b are odd.

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Answers (1)

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Elisa · Answer 1

Yes!

Step-by-step explanation:

Let x be the tens place and y be the units place. x and y need to be odd because in other case a or b will not be odd. For example, if x=1 and y=2 a will be 21 but b will be 12 that is not odd.

Now, a+b = xy+yx. Note that xy is not x*y, is just the digits concatenated. Then, there are two cases:

If x+y<10 then xy+yx = (x+y) (x+y) (again, that is not a multiplication is x+y concatenated with x+y) and that is 11 * (x+y) a multiple of 11.

If x+y≥10 then xy+yx = (x+y+1) (x+y-10) because x+y<20.

Now we are going to see that last result without the concatenation, as a sum, that is

(x+y+1) * 10 + x+y-10 = 10x+10y+10+x+y-10 = 11x+11y = 11 (x+y). This result is clearly a multiple of 11.

In conclusion, in all cases the result of a+b is a multiple of 11.