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Subject: "10 digits add to 100"     Previous Topic | Next Topic
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Conferences The CTK Exchange High school Topic #405
Reading Topic #405
huntermykey
Member since Feb-23-11
Feb-23-11, 01:18 PM (EST)
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"10 digits add to 100"
 
   Hi everyone,

I am not sure if i am plane stupid,( probably am) or this is big Math..

This is the problem:
Use all digits 0-9 once, in no specific order or combination, so they add up to 100.

So ONLY + allowed

Add hint, there is one prime number

Maybe, just maybe think out of the box, like binairi???
Stuck for real,

My'key


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alexbadmin
Charter Member
2772 posts
Feb-23-11, 02:05 PM (EST)
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1. "RE: 10 digits add to 100"
In response to message #0
 
   I do not think this is possible.

Proof

0 + 1 + 2 + 3 + ... + 8 + 9 = 45

In the sum we are looking for there are only 1- and 2-digit numbers.

Assume there is a decimal number ab, i.e. 10a + b.

The presence of ab modifies the sum above, i.e., 45, by

10a + b - a - b = 9a

which is a factor of 9. But 45 itself is divisible by 9, implying that any sum in which all 10 digits appear joined in at most a 2-digit number is always divisible by 9.

Actually this is clear from the fact that "the sum of digital roots is the digital root of the sum."


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