### Today’s Puzzle:

Do most of the numbers from 1501 to 1600 have 2 factors, 4 factors, 6 factors, or what? A horse race is a fun way to find the answer to that puzzle!

As I’ve done several times before, I’ve made a horse race for this multiple of 100 and the 99 numbers before it. A horse moves when a number comes up with a particular amount of factors. Some of the races I’ve done in the past have been exciting with several lead changes. In other races, one horse ran quite quickly, leaving all other horses in the dust. One previous horse race resulted in a tie.

How will you describe this horse race? Exciting or boring? Surprizing or predictable? Pick your pony and watch the race to the end before you decide on an adjective.

Click here if you would like the Horse Race to be slightly bigger.

### Prime Factorization of Numbers from 1501 to 1600:

Of those 100 numbers, 38 have square roots that can be simplified; 62 do not.

### Factor Trees for 1600:

1600 has MANY possible factor trees. Some are symmetrical; some are not. Here are two nicely shaped ones:

### Factors of 1600:

- 1600 is a composite number and a perfect square.
- Prime factorization: 1600 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5, which can be written 1600 = 2⁶ × 5².
- 1600 has at least one exponent greater than 1 in its prime factorization so √1600 can be simplified. Taking the factor pair from the factor pair table below with the same number for both factors, we get

√1600 = (√40)(√40) = 40. However, you could also use

√1600 = (√4)(√400) = 2 × 20 = 40,

√1600 = (√16)(√100) = 4 × 10 = 40, or

√1600 = (√25)(√64) = 5 × 8 = 40. - The exponents in the prime factorization are 6 and 2. Adding one to each exponent and multiplying we get (6 + 1)(2 + 1) = 7 × 3 = 21. Therefore 1600 has exactly 21 factors.
- The factors of 1600 are outlined with their factor pair partners in the graphic below.

### More about the Number 1600:

1600 is the sum of two squares:

32² + 24² = 1600.

1600 is the hypotenuse of two Pythagorean triples:

448-1536-1600, calculated from 32² – 24², 2(32)(24), 32² + 24².

It is also (7-24-**25**) times **64**.

960-1280-1600, which is (3-4-**5**) times **320**.

1600 looks square in some other bases:

1600 = 1(40²) + 0(40) + 0(1), so it’s 100₄₀.

1600 =1(39²) + 2(39) + 1(1), so it’s 121₃₉.

1600 =1(38²) + 4(38) + 4(1), so it’s 144₃₈.

1600 =1(37²) + 6(37) + 9(1), so it’s 169₃₇.

Furthermore, 1600 is a repdigit in base 7:

1600 = 4(7³ + 7² + 7¹ + 7º), so it’s 4444₇.