### Today’s Puzzle:

How can you arrange 1513 dots into a perfect square when √1513 is irrational?

The answer is you arrange the dots into a centered square like this:

You can arrange them like that because 1513 is the sum of consecutive squares.

### Factors of 1513:

- 1513 is a composite number.
- Prime factorization: 1513 = 17 × 89
- 1513 has no exponents greater than 1 in its prime factorization, so √1513 cannot be simplified.
- The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1513 has exactly 4 factors.
- The factors of 1513 are outlined with their factor pair partners in the graphic below.

### More about the Number 1513:

1513 is the sum of two squares in two different ways:

28² + 27² = 1513, and

37² + 12² = 1513.

1513 is the hypotenuse of FOUR Pythagorean triples:

55-1512-1513, calculated from 28² – 27², 2(28)( 27), 28² + 27²

663-1360-1513, which is **17** times (39-80-**89**)

712-1335-1513, which is (8-15-**17**) times **89**

888-1225-1513, calculated from 2(37)(12), 37² – 12², 37² + 12²

### Could 1513 be a prime number?

Since its last two digits divided by 4 leave a remainder of 1, and 28² + 27² = 1513 with 28 and 27 having no common prime factors, 1513 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1513. Is 1513 divisible by 5, 13, 17, 29, or 37? Yes, it is divisible by 17, so 1513 is NOT a prime number.

37² + 12² = 1513 and 37 and 12 have no common prime factors, so we could have arrived at the same result using those numbers.

Note: Numbers that are the sum of two squares in two or more ways are never prime.