Number of solutions
Find the number of solutions in positive integers of the equation 3 x + 5 y = 1 0 0 8 .
The answer is 67.
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2 solutions
good one!! see if you can understand my solution.
Let
x
,
y
be natural numbers such that
3
x
+
5
y
=
1
0
0
8
; (let
|
be the symbol for divides)
then
3
∣
5
y
⇒
3
∣
y
⇒
y
=
3
k
; for some
k
belonging to a natural number.Now
3
x
+
1
5
k
=
1
0
0
8
⇒
x
+
5
k
=
3
3
6
⇒
5
k
≤
3
3
5
⇒
k
≤
6
7
.
Thus any solution pair is given by
(
x
,
y
)
=
(
3
3
6
−
5
k
,
3
k
)
where
1
≤
k
≤
6
7
and therefore the number of solutions is
6
7
.
Clear detailed explanation... +1
Well we know that the numbers have to add up to 1003, and 1 must be divisible by 3, while the other is divisible by 5.
LCM (lowest common multiple)
The LCM of 3 and 5 is 15. Therefore, every multiple of 15 from 1 - 1003, will tell us how many possibilities there is.
1003 / 15 = 67 remainder 3
Therefore, our answer is 67