Question

find 2 integers such that they sum to: 1640

and their LCM is: 8400

Answer 1

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

Answer 2

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

Answer 3

Maybe algebra with a quadratic equation?

Answer 4

maybe. but number theory methods might be perfered

Answer 5

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.

Answer 6

\[\large x \left( 1640-x \right)=8400n\]

where n is a natural number

Answer 7

hmm, the "n" seems interesting

Answer 8

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

Answer 9

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

Answer 10

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

Answer 11

needs to be a perfect square

Answer 12

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

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

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},