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# Representation of numbers in base 10 AMC 8 2013 , problem number 13

Try this beautiful problem from AMC 8. It involves representation of numbers in base 10. We provide sequential hints so that you can try the problem.

# What are we learning ?

Competency in Focus: Representation of numbers in base 10 This problem from American Mathematics contest (AMC 8, 2013) is a digit problem .

# First look at the knowledge graph. # Next understand the problem

When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?
##### Source of the problem
American Mathematical Contest 2013, AMC 8 Problem 13
##### Key Competency
Representation of numbers in base 10
4/10
##### Suggested Book
Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics

Do you really need a hint ? Try it first!
Let the two digits be $a$ and $b$ that is total score of Clara is ab .
Now we know that any number in base 10 can be represented as ab=10 a+ b.  Given, when Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. So, Clara misinterpreted ab as ba . Again , ba= 10b+a .
The difference between the two is $|9a-9b|$ which factors into $|9(a-b)|$. So, what can we say from here?
Therefore, since the difference is a multiple of 9, the only answer choice that is a multiple of 9 that is 45.

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