Triple ABC

Triple ABC
How many ordered triples \left (A,B,C\right ) of positive integers satisfy ABC = 4000?
Source: mathcontest.olemiss.edu 9/10/2012

SOLUTION
This problem is similar to the problem “XYZ=4000” posted on 11/5/2014, nevertheless a different solution is offered here.

Let N be a positive integer and N=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k} its prime factorization, then the number of divisors of N equals \left (e_1+1\right )\left (e_2+1\right )\cdots\left (e_k+1\right ). For example,
N=18=2^1\times 3^2
Number of divisors
\left (1+1\right )\left (2+1\right )=6
Divisors
1,2,3,6,9,18

N=4000=2^5\times 5^3
Number of divisors
\left (5+1\right )\left (3+1\right )=24
Divisors
R\!:1,2,4,8,16,32
S\!:5,10,20,40,80,160
T\!:25,50,100,200,400,800
U\!:125,250,500,1000,2000,4000

We multiply the divisors to find the triples \left (A,B,C\right ) and  use multiplication tables to show the products.
R\times U multiplication table

The 4000 diagonal gives the \left (1,B,C\right ) triples; the 2000 diagonal gives the \left (2,B,C\right ) triples; the 1000 diagonal gives the \left (4,B,C\right ) triples, and so on. Beware that there will be duplicates. For simplicity, we list the triples in increasing numerical order.
R\times U yields 12 distinct triples
\left (1,1,4000\right )\quad\left (2,2,1000\right )\quad\left (4,4,250\right )
\left (1,2,2000\right )\quad\left (2,4,500\right )\quad\;\;\left (4,8,125\right )
\left (1,4,1000\right )\quad\left (2,8,250\right )
\left (1,8,500\right )\quad\;\;\left (2,16,125\right )
\left (1,16,250\right )
\left (1,32,125\right )

S\times T multiplication table

S\times T yields 21 distinct triples
\left (1,5,800\right )\quad\;\;\left (2,5,400\right )\quad\;\;\left (4,5,200\right )\quad\;\left (5,8,100\right )\quad\left (8,10,50\right )\quad\left (10,16,25\right )
\left (1,10,400\right )\quad\left (2,10,200\right )\quad\left (4,10,100\right )\quad\left (5,16,50\right )\quad\left (8,20,25\right )
\left (1,20,200\right )\quad\left (2,20,100\right )\quad\left (4,20,50\right )\quad\;\;\left (5,25,32\right )
\left (1,25,160\right )\quad\left (2,25,80\right )\quad\;\;\left (4,25,40\right )
\left (1,40,100\right )\quad\left (2,40,50\right )
\left (1,50,80\right )

R\times S yields duplicate triples
R\times T yields duplicate triples
S\times U yields no solution
T\times U yields no solution
R\times R yields no solution
T\times T yields no solution
U\times U yields no solution

S\times S multiplication table

S\times S yields 5 distinct triples
\left (5,5,160\right )\quad\left (10,10,40\right )
\left (5,10,80\right )\quad\left (10,20,20\right )
\left (5,20,40\right )

The complete list of the 38 distinct triples is shown below
The problem asks for ordered triples.
\left (1,1,4000\right )\!,\left (2,2,1000\right )\!,\left (4,4,250\right )\!,\left (5,5,160\right )\!,\left (10,10,40\right )\!,\left (10,20,20\right ) give
6\times 3=18 ordered triples.
The remaining 32 triples give
32\times 6=192 ordered triples.
Total number of ordered triples \left (A,B,C\right ) such that ABC=4000
18+192=210

Answer: 210

Advertisements

About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , , , , , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s