"Mr Macaw" wrote in news: snipped-for-privacy@red.lan:
I'll illustrate the principle using a 3-digit number. Pick any three digits, and designate them as a, b, and c.
The sum of the digits is a + b + c; call this s. The value of the number is (100 * a) + (10 * b) + c; call this v.
The difference between the value of the number, and the sum of its digits, is v - s = (100a + 10b + c) - (a + b + c) = 99a - 9b = 9(11a - b)
Therefore, (v - s) must be a multiple of 9 (and hence of 3 also).
If s is a multiple of 3, and you add a multiple of 3 to it, the result must also be a multiple of 3, and likewise for multiples of 9.
Therefore if s is a multiple of 3 (or 9) then v = s + (v - s) is also a multiple of 3 (or 9).
For a four-digit number abcd, (v - s) = 999a + 99b + 9c = 9(111a + 11b + c) which is clearly a multiple of 9. And so on, for numbers having 5, 6, 7, etc. digits.
Yes. If someone says "12 is divisible by 4", they mean "to produce a wh= ole number". If someone said "13 is divisible by 4" they would be incor= rect.
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If you own a =A33,000 machine gun and a =A35,000 rocket launcher, but yo= u can't afford shoes, you may be a Muslim.
Your two references contradict each other. One says that just divisible means it has to be evenly divisible, ie with no remainder. That seems to be the correct definition, multiple source confirm it. So saying evenly divisible is not needed, it's already implied.
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