Let $\displaystyle f(n)$ denote the sum of the digits of the positive integer $\displaystyle n$ in decimal notation. What may $\displaystyle f(3a)$ be if $\displaystyle f(a)=100$ and $\displaystyle f(124a)=700$?

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- Dec 15th 2007, 02:32 AM #1

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- Dec 15th 2007, 03:35 AM #2

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Hello, james_bond!

Is this a trick question?

Let $\displaystyle f(n)$ denote the sum of the digits of the positive integer $\displaystyle n$ in decimal notation.

What may $\displaystyle f(3a)$ be if $\displaystyle f(a)=100$ and $\displaystyle f(124a)=700$?

Anyone familiar with "Casting out 9's" know this fact . . .

. . If $\displaystyle f(n)$ is the digital-sum function: .$\displaystyle f(a\cdot b) \:=\:f(a)\cdot f(b)$

(Hmm, that's not quite accurate, but it'll do.)

If $\displaystyle f(a) = 100$, then: .$\displaystyle f(3a) \;=\;f(3)\!\cdot\!f(a) \;=\;3\cdot100 \;=\;300 $ .*. . . too easy*

- Dec 15th 2007, 04:20 AM #3

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- Dec 21st 2007, 06:07 PM #4
I know.

*a*is the 100-digit number consisting of all 1’s, so 124*a*is the 102-digit number whose first two digits are 13 and last two 64, with 98 7’s in between. Hence f(3*a*) = 300 after all.

- Dec 22nd 2007, 04:42 AM #5

- Dec 22nd 2007, 05:30 AM #6