## amistre64 Group Title find 2 integers such that they sum to: 1640 and their LCM is: 8400 one year ago one year ago

1. amistre64 Group Title

my idea is to factor the LCM to create a pool of options

2. amistre64 Group Title

8400 = 84*100 = 84*2*2*5*5 = 2*2*3*7*2*2*5*5 2*2*2*2*3*5*5*7, therefore 2 2*2 2*2*2 2*2*2*2 2*2*2*2*3 2*2*2*2*3*5 2*2*2*2*3*5*5 2*2*2*2*3*5*5*7 etc ... but that does seem like a long way around it

3. CliffSedge Group Title

Maybe algebra with a quadratic equation?

4. amistre64 Group Title

maybe. but number theory methods might be perfered

5. CliffSedge Group Title

Might want to use optimization methods from calculus too to minimize the coefficients. Yes, it does seem like a number theory issue, but algebra is always my starting point for solving unknowns.

6. PaxPolaris Group Title

$\large x \left( 1640-x \right)=8400n$ where n is a natural number

7. amistre64 Group Title

hmm, the "n" seems interesting

8. amistre64 Group Title

on the test i just ended up using the y= on my ti83 y1 = 1640-x y2= lcm(x,1640-x) then searched thru that table till i found y2 = 8400

9. PaxPolaris Group Title

$x=820 \pm \sqrt {820^2-8400n}$

10. amistre64 Group Title

i assume the under radical needs to remain 0 or greater?

11. PaxPolaris Group Title

needs to be a perfect square

12. amistre64 Group Title

lol, yeah i spose that would have to be a major caveat seeing the x needs to be an integer :)

13. PaxPolaris Group Title

$\large x=820 \pm 400\sqrt{1681-21n}$

14. amistre64 Group Title

i think one more condition was such that x and y were both positive values

15. PaxPolaris Group Title

sorry, $\large x=820 \pm \sqrt{400(1681-21n)}$$\large x=820 \pm 20\sqrt{1681-21n}$ and n is the GCF of x and y

16. robtobey Group Title

240 + 1400 = 1640, LCM[240, 1400] = 8400 Used the following small program and fed it to Mathematica. Table[{n, LCM[n, 1640 - n] == 8400}, {n, 820}] Part of the Output: {233, False}, {234, False}, {235, False}, {236, False}, {237, False}, \ {238, False}, {239, False}, {240, True}, {241, False}, {242, False}, \ {243, False}, {244, False},