Multiples of Nine

[Brutal]

5 x 9 = 45. 4 + 5 = 9 why? 77 x 9 = 693. 6 + 9 + 3 = 18 (multiple of 9) why? This works with any number multiply by 9. why?

[Dsg]

Take the simple case for 9x where x is a number less than 10. 9x = 10x-x = 10(x-1) + (10-x) So the first digit (the tens) of the answer is (x-1) and the second digit (the units) is (10-x) - therefore whatever the value of x:- (x-1)+(10-x) = 9. The same can be demonstrated for the case for x being greater than 10.

[Bert]

The first answer can hardly be bettered, but I'd just like to add that this 'phenomenon' is the basis for the test for the divisiblity of any number by 9. If the sum of the digits of a number is divisible by 9, then the number itself is divisible by 9. There is also a way of checking additions by "casting out nines" but I have forgotten how that works.