### Today’s Puzzle:

Sure, it’s a rectangle with whole-number sides in 10 different ways, but what kind of REGULAR polygonal shape can 1520 be made into? I will tell you that the measurement of each of its sides is 32.

And thus, it is the 32nd shape of its kind. By the way, I really like how all the 32nd figurate numbers relate to each other:

We see in the chart that 1520 dots can be arranged into a pentagon. Just how do we do that? Here’s how:

Do you see from the graphic that 1520 is 32 more than three times the 31st triangular number?

1520 is also related to triangular numbers in another way: Today I learned that all pentagonal numbers are 1/3 of a triangular number. Indeed, 1520 is 1/3 of the 95th triangular number:

(1/3) of (95)(96)/2 = 1520.

Pretty cool, I think!

### Factors of 1520:

- 1520 is a composite number.
- Prime factorization: 1520 = 2 × 2 × 2 × 2 × 5 × 19, which can be written 1520 = 2⁴ × 5 × 19.
- 1520 has at least one exponent greater than 1 in its prime factorization so √1520 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1520 = (√16)(√95) = 4√95.
- The exponents in the prime factorization are 4, 1, and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 1520 has exactly 20 factors.
- The factors of 1520 are outlined with their factor pair partners in the graphic below.

### More about the Number 1520:

Recently someone on twitter asked:

9 times 9 is 81, and 10 times 8 is 80.

12 times 12 is 144, and 13 times 11 is 143.

5 times 5 is 25, and 6 times 4 is 24.20 times 20 is 400, so what’s 21 times 19?

can you explain why this is happening, without using algebra? (i mean, go ahead with the algebra, if you insist)

— Laurie Rubel (@Laurie_Rubel) September 6, 2020

If you look at the whole thread, you will see how a few people explained this important concept using arrays. Here is my attempt to explain the difference of two squares using 1520 and arrays:

As I mentioned before, 1520 has 10 rectangles with whole-number sides. The one with the smallest perimeter is 38 × 40, and it is the easiest to use to demonstrate how 1520 is the difference of two squares:

I made that gif be as slow as I could without duplicating any of the frames, but it still goes pretty fast.

1520 is, in fact, the difference of two squares in six different ways:

39² – 1² = 1520,

48² – 28² = 1520,

81² – 71² = 1520,

99² – 91² = 1520,

192² – 188² = 1520, and

381² – 379² = 1520.

1520 is also the hypotenuse of a Pythagorean triple:

912-1216-1520, which is (3-4-**5**) times **304**.

From OEIS.org we learn that 1520 and 1521 are a Ruth-Aaron pair because they are consecutive numbers and the sum of their factors are equal to each other:

1520 = 2⁴·5·19 and 1521 = 3²·13², while

2+2+2+2+5+19 = 32=3+3+13+13.