Find positive integers x, y, satisfying x^2+y^2=6625. Mathematics competition | Mathematics Olympiad

The video illustrates step-by-step how to find positive integers x, y, which satisfies x^2+y^2=6625. In general, to find these positive integers, it’s useful to narrow down the range of possible integers (e.g. by checking the parity of the variables), especially for larger numbers.

However, an effective method to solve this kind of problems may vary from problem to problem. For this problem, noticing 6625 congruent to 1 (mod 3) is vital to solving the problem.

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[mathsenhancersclass]

Update to video (around 13:50):
“When t=2, 3, 4, 5, 7, 8, there are no solutions for (2).” should be
“When t=2, 3, 4, 5, 7, there are no solutions for (2).”
For t=8, (2) can be written as m^2+9∙64±16=736.
m^2=176 (discarded), or m^2=144, m=12.
So x=3m=3∙12=36, y=3n+1=9t+1=9∙8+1=73.
So (x, y)=(36, 73), (73, 36) also satisfy the equation.

[rachid]

Dans le cas m^2+9t^2-2t=736 comment vous avez conclu rapidement que t est inférieur à 9 ?même si c'est vrai, les justifications sont à revoir .

[sy]

For the additional problem: My answer is (u, v) = (44, 8), (40, 20), (20, 40), (8, 44) [How to solve is almost same as the past problem a few days ago.]

2000 can be divided by 4. (odd number)^2+(even number)^2 cannot be any even number.
where k, k': integer This means (odd number)^2+(odd number)^2 cannot be divided by 4.
Therefore, both u and v should be even numbers. Set u=2m, v=2n. Then (2m)^2+(2n)^2=2000 . ∴ 4m^2+4n^2=2000 ∴ m^2+n^2=500
4 | 500 By using same theory, both m and n should be even numbers. Set m=2s, n=2t. Then (2s)^2+(2t)^2=500 .
∴ 4s^2+4t^2=500 ∴ s^2+t^2=125 ∴ u, v satisfy "u=4s, v=4t, s^2+t^2=125"
By the way, 12^2=144>125, 11^2=121<125, and 8^2=64>125/2(=62.5), 7^2=49<125/2(=62.5)
From these, if set s>t, candidate solutions of s are 8, 9, 10, and 11.
If s=8, t^2=125-s^2=125-64=61 (not perfect square)
If s=9, t^2=125-s^2=125-81=44 (not perfect square)
If s=10, t^2=125-s^2=125-100=25 ∴ t=5 (∵ t>0)
If s=11, t^2=125-s^2=125-121=4 ∴ t=2 (∵ t>0)
∴ (s, t)=(10, 5), (11, 2) ∴ (u, v) = (4s, 4t) = (40, 20), (44, 4) Since u and v are symmetrical, (u, v) = (44, 8), (40, 20), (20, 40), (8, 44)
[Check: ] 40^2+20^2=1600+400=2000, 44^2+8^2=1936+64=2000 (Both are OK.)
Is my method (and calculation) right?

Appendix: Yesterday, I commented that I made math problems and uploaded. I commented yesterday as follows:
[ Today's problem in this video is an integer problem, so I introduced my videos which has a lot of integer problems. ]

[rachid]

In the case m^2+9t^2-2t=736 how did you quickly conclude that t is less than 9? even if it is true, the justifications are to be reviewed.