Countdown to 2014: Problem #1

Number Theory Level 3

How many values of x x are there, such that x x and 2014 x \frac{2014}{x} are both positive integers?


The answer is 8.

Wee Xian Bin by Wee Xian Bin , 25, Singapore

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

3 solutions

Isaac Lu
Mar 5, 2014

Prime factors of 2014 are 2, 19, and 53.

Two factors: x and 2014/x.

We find the number of ways to combine two factors. P(3,2)=6

Also include that 1 and 2014 are factors of 2014. 1 and 2014 is different from 2014 and 1. Add these two possibilities to 6 possibilities.

We got 8. There are 8 values for x.

1 and 2014, 2 and 1007, 19 and 106, 38 and 53, 53 and 38, 106 and 19, 1007 and 2, and 2014 and 1.

4 Helpful 0 Interesting 0 Brilliant 0 Confused

Very good solution, good reasoning. I feel good with this answer...

Heder Oliveira Dias - 6 years, 5 months ago
Rama Devi
May 16, 2015

The solution is nothing but the number of factors of 2014 , that is 8.

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Vishnudatt Gupta
May 13, 2014

2014 = 2 x 19 x 53*1

so 1,2,19,53 are solutions

now 2 19,2 53,19*53 will be also the soluiton

so total=8

1 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...