65 is a composite number. 65 = 1 x 65 or 5 x 13. Factors of 65: 1, 5, 13, 65. Prime factorization: 65 = 5 x 13.

65 – 56 = 9, a square number and 65+56 = 121, another square number. Stetson.edu informs us that 65 and 56 are the smallest pair of mirror image numbers that produce a square number when either added or subtracted.

65 = (1 + 2 + 3 +. . . + 23 + 24 + 25)/5 so it is the magic sum of a 5 × 5 magic square. I solved this 5×5 magic square game using a time honored algorithm. Notice that 11, 12, 13, 14, and 15 are on a diagonal. Actually all of the numbers are sort of on a diagonal if you follow the instructions given here. Also notice that the first number, 1, is in the center of the top row and the last number, 25, is in the center of the bottom row.

You can try to solve it, too. You will know you have solved it when the sum of each row, column, and diagonal totals 65.

65 is also the sum of two squares two different ways:

- 8² + 1² = 65
- 7² + 4² = 65

That makes 65 the hypotenuse of FOUR Pythagorean triples:

- 16-63-65 which was calculated from 2(8)(1), 8² – 1², 8² + 1²
- 25-60-65 which is 5 times 5-12-13
- 33-56-65 which was calculated from 7² – 4², 2(7)(4), 7² + 4²
- 39-52-65 which is 13 times 3-4-5

Comments on:"65 is the Magic Sum of a 5 x 5 Magic Square" (2)Joseph Nebussaid:Ooh, congratulations! I imagine that’s got to be most gratifying to see.

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ivasallaysaid:Yes, and I think their teachers are extraordinary because they expose the students to many fun mathematics ideas, (and they teach the students in both Spanish and English!)

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