The difference of the squares of two numbers is 45. The squares of the smaller number is 4 times the larger number. Determine the numbers?

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What is a basis in linear algebra?
There are many different bases for a vector space, but all bases for the same vector space have the same number of vectors. This number is called the dimension of the vector space
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Find three numbers in A.P. whose sum is 21 and their product is 231.
Let, three numbers are (a-d), a, (a+d) So, (a-d)+a+(a+d)=21 => a=7 Again, (7-d)×7×(7+d)=231 => (7-d)×(7+d)=231÷7 => 7^2 - d^2 = 33=> d^2 = 49 - 33=> d^2 = 16=> d = 4 So, the numbers are 7-4= 3, 7 and 7+4=11
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If m and n are odd positive integers, then m2 + n2 is even, but not divisible by 4. Justify.
Since m is odd positive integer then we can write m=2a+1 and n=2b+1, such that a,b >0 m2+n2=(2a+1)^2+(2b+1)^2 = 4a^2+1+4a+4b^2+1+4b = 4a^2+4b^2+4a+4b+2 = 2(2a^2+2b^2+2a+2b+1) Above eqn is divisible by 2 but not divisible by 4.
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