Question

express the square number 225 as the sum of two consecutive natural numbers​

Answer 1

Use your results to find the squares that can be added to 225 to produce another square.

I started off by taking the 9 divides 225 with quotient 25.

(25-8) + (25-6) + (25-4) + (25-2) + 25 + (25+2) + (25+4) + (25+6) + (25+8) = 225

simplifying:

17 + 19 + 21 + 23 + 25 + 27 + 29 + 31 + 33 = 225.

Since integers can be negative, some ways will start with negative integers. For example 45 consecutive odd integers can add to 225 like this: (-39) + (-37) + (-35) + ... + 45 + 47 + 49 = 225.

So, to answer your question you have to count the number positive odd divisors of 225, and that will be the cardinality of the the set {1, 3, 5, 9, 15, 25, 45, 75, 225}, so the answer is 9.