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Prove algebraically that the sum of the squares of any 2 even positive integers is always a multiple of 4.

(chatgpt is not helping, my tutor is asleep)

help this is my last resort

6 7

The sum of the squares of any two even positive integers is proven to be a multiple of 4 by representing the integers as 2n and 2m, squaring them, and showing that their sum can be factored out as 4 times another integer.

https://www.quora.com/Prove-that-the-sum-of-the-squares-of-any-two-consecutive-even-numbers-is-always-a-multiple-of-4

Let 2 even numbers be 2x and 2y and so sum of their squares is 4x^2+4y^2 which is a multiple of 4

Bro what grade are u in don't use chatgpt for this or u won't be able live in real world

Man how can I send you picture, I can do it on paper but typing allat

Im not a math expert so I asked chatgpt
(2m)² + (2n)² = 4m² + 4n² = 4(m² + n²)
Let the two even positive integers be 2m and 2n, where m, n are positive integers
Since m and n are integers, (m² + n²) is also an integer, so the result is 4 × (an integer), which is by definition a multiple of 4

tubelight [#3]

Let 2 even numbers be 2x and 2y and so sum of their squares is 4x^2+4y^2 which is a multiple of 4

Bro what grade are u in don't use chatgpt for this or u won't be able live in real world

grade 9, im lwk cooked, cant it just be x and y?

for two positive even integers 2a and 2b

(2a)^2 = 4a^2 and (2b)^2 = 4b^2

(2a)^2 + (2b)^2 = 4a^2 + 4b^2

(2a)^2 + (2b)^2 = 4(a^2 + b^2)

(a^2 + b^2) is an integer so the sum of the squares of two positive even integers will always be a multiple of 4

6+7=21

meth expert here

(6 + 7)^2= (67)^2
im an engineer btw

cameronto09 [#6]

grade 9, im lwk cooked, cant it just be x and y?

It is just x and y, however, since you said both of them are even numbers, that means both of them are divisible by 2 i.e. x/2 would give an integer; similarly y/2 will also give an integer since y is divisible by 2.
Say, x/2 = a (an integer)
=> x = 2a
Similarly, y/2 = b (an integer)
=> y = 2b
Now, x^2 + y^2 = (2a)^2 + (2b)^2
= 4(a^2) + 4(b^2)
= 4 (a^2 + b^2)

cameronto09 [#6]

grade 9, im lwk cooked, cant it just be x and y?

It can be but you need to use variables according to the question.
Every even number is a multiple of 2 so even numbers can be anything like 2 times x,y,z,a,b,c...
For this question when you use 2x or 2y whatever, their squares will be multiples of 4 and so, the sum will be also a multiple of 4.

tubelight [#11]

It can be but you need to use variables according to the question.
Every even number is a multiple of 2 so even numbers can be anything like 2 times x,y,z,a,b,c...
For this question when you use 2x or 2y whatever, their squares will be multiples of 4 and so, the sum will be also a multiple of 4.

ah ok, its just i havent seen proof questions before

..... you wont survive calc