Find the smallest multiple of 17 that has only 1s in its decimal expansion.

Hint: the numbers in question are of the form (10^n-1)/9.

How should I solve this question? Please teach me . Thank you.

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- Dec 25th 2006, 05:25 AMJenny20repunits
Find the smallest multiple of 17 that has only 1s in its decimal expansion.

Hint: the numbers in question are of the form (10^n-1)/9.

How should I solve this question? Please teach me . Thank you. - Dec 25th 2006, 06:36 AMSoroban
Hello, Jenny!

Quote:

Find the smallest multiple of 17 that has only 1's in its decimal expansion.

Hint: the numbers in question are of the form (10^n - 1)/9

I haven't found a way to use that hint.

If you're desperate, here's a**very**primitive method.

Divide 17 into a "string of 1's" ... and wait for it to*come out even*.Code:

6 5 3 5 9 4 7 7 1 2 4 1 8 3

---------------------------------

1 7 ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

--------------------------------

0

Therefore: 1,111,111,111,111,111 is the smallest multiple of 17.

- Dec 25th 2006, 07:29 AMJenny20
Hi Soroban,

Thank you very much. I hope that I can use the hint to solve this question.

My friend came up this :

17n = (10^n-1)/9

Is it useful regarding to the hint? - Dec 25th 2006, 07:58 AMThePerfectHacker
I think you mean,

17m=(10^n-1)/9

The number on left is multiple of 17.

The number on right is a*repunit*.

Multiply by 9,

153m=10^n-1

Thus,

153m+1=10^n

Note, the expression on right is 1 followed by zeros.

Thus, find the smallest "m" that will make,

153m+1

Be followed by zeros. - Dec 25th 2006, 08:10 AMJenny20
Hi perfecthacker,

Can you use the " hint " to show me the way to solve my question? If possible, please do show me the working steps. Thank you very much. - Dec 25th 2006, 08:51 AMCaptainBlack
You can combine the hint with soroban's method, as:

17.N=(10^n-1)/9

means:

153.N=10^n-1,

but the rigt hand side is a number all of whoes digits are 9, so you can do

the long division with 153 instead of 17 and a string of 9's instead of 1's

in soroban's method

RonL