Difference between revisions of "AoPS Wiki talk:Problem of the Day/June 12, 2011"
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{{:AoPSWiki:Problem of the Day/June 12, 2011}} | {{:AoPSWiki:Problem of the Day/June 12, 2011}} | ||
==Solution== | ==Solution== | ||
| − | {{ | + | Let's take a look at the units digit of <math>39</math>, which is <math>9</math>. Now, let's take a look at the positive numbers that add up to to <math>9</math>: |
| + | |||
| + | <cmath>1,8</cmath> | ||
| + | |||
| + | <cmath>2,7</cmath> | ||
| + | |||
| + | <cmath>3,6</cmath> | ||
| + | |||
| + | <cmath>4,5</cmath> | ||
| + | |||
| + | Now, we realize that every pair has an even element. Any number with an even units digit is even. So, what is the only even prime? <math>2</math>! So, one of our primes is <math>2</math>. The other is then, consequently, <math>37</math>. The product is then: | ||
| + | |||
| + | <cmath>2\cdot{37}=\boxed{74}</cmath> | ||
Latest revision as of 17:40, 11 June 2011
Problem
AoPSWiki:Problem of the Day/June 12, 2011
Solution
Let's take a look at the units digit of 
, which is 
. Now, let's take a look at the positive numbers that add up to to :
![\[1,8\]](http://latex.artofproblemsolving.com/e/7/2/e729dde3f194f2020f443e147993afd6df22bcab.png)
![\[2,7\]](http://latex.artofproblemsolving.com/4/6/a/46ac7db3eee123eadc90e7ca58497db1e64ad702.png)
![\[3,6\]](http://latex.artofproblemsolving.com/d/7/c/d7c9016b28e37792c708a06a5f2c27a25108966b.png)
![\[4,5\]](http://latex.artofproblemsolving.com/c/d/3/cd340c60c427b0a23e1930aafeb8ae3f5b0e981f.png)
Now, we realize that every pair has an even element. Any number with an even units digit is even. So, what is the only even prime? 
! So, one of our primes is . The other is then, consequently, 
. The product is then:
![\[2\cdot{37}=\boxed{74}\]](http://latex.artofproblemsolving.com/e/1/c/e1cdf017ccc5841ed8644feed5d4c1641596d43f.png)
