Divides by 9?

Number Theory Level pending

What number would need to be added to the following number to make it divisible by 9?

( 12345678987654321 ) 12 \large (12345678987654321)_{12}

Clarification : The subscript indicates base 12.

Hint : There is no need to convert the above number to decimal or use a calculator. Good luck!


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1 solution

Geoff Pilling
May 3, 2016

Since ( 1 2 10 ) 2 (12_{10})^2 divides evenly by 9 9 , you need only look at the last two digits. In base 12 12 , 2 1 12 21_{12} doesn't divide by 9 9 , but 2 3 12 23_{12} does. ( 2 12 + 3 ) 10 = ( 27 ) 10 (2*12+3)_{10} = (27)_{10} . Therefore, you would need to add 2 \boxed2 to this number to make it divisible by 9 9 .

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