Factors and Facts for the Year 2021

Countdown to 2021:

Here is a countdown you can use on New Year’s Eve to bring in 2021:

Countdown to 2021

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Math Facts about the Number 2021:

2021 Tiny Squares

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2021 is the difference of two squares in two different ways:
45² – 2² = 2021 and
1011² – 1010² = 2021.

2021 is not the hypotenuse of any Pythagorean triples, but it is a leg in four:
180-2021-2029, calculated from 2(45)(2), 45² – 2², 45² + 2²,
2021-43428-43475, which is (43-924-925) times 47,
2021-47472-47515, which is 43 times (47-1104-1105)
2021-2042220-2042221, from 1011² – 1010², 2(1011)(1010), 1011² + 1010².

2021 is one of the numbers in the table below because
2021² = 4084441, and their reverses are 1202² = 1444804.
For each pair of numbers in the table, the smaller number is listed first, but if you look carefully you will notice 2001, 2011, 2012, 2021, and 2022 in this century and 2101, 2102, 2111, and 2121 in the next.

Factors of 2021:

The previous year had many negative factors, although it had just as many positive ones. I can predict that 2021 will have only four negative factors to go along with its four positive ones. I think we will see a lot of the negative factors at the beginning of the year, but hopefully, the positive ones will be more evident as the year goes on.

  • 2021 is a composite number.
  • Prime factorization: 2021 = 43 × 47.
  • 2021 has no exponents greater than 1 in its prime factorization, so √2021 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 2021 has exactly 4 factors.
  • The factors of 2021 are outlined with their factor pair partners in the graphic below.

Those factors can help you with quadratic equations like these:

Mathematical Tweets about 2021:

Here are tweets about the number 2021 in the order that I’ve seen them. If I see more, I’ll add more.

 

1556 Stacks Up Nicely!

What is special about the number 1556?

What makes 1556 stack up?

From OEIS.org we learn that
2² + 3² + 5² + 7² + 11² + 13² + 17² + 19² + 23² =  1556.
Yes, that’s the sum of the squares of the first nine prime numbers.
Those perfect squares can be stacked on top of each other as I illustrate in the graphic below:

Factors of 1556:

1556 (and every other whole number whose last two digits are 56) is divisible by 4:

  • 1556 is a composite number.
  • Prime factorization: 1556 = 2 × 2 × 389, which can be written 1556 = 2² × 389
  • 1556 has at least one exponent greater than 1 in its prime factorization so √1556 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1556 = (√4)(√389) = 2√389
  • The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1556 has exactly 6 factors.
  • The factors of 1556 are outlined with their factor pair partners in the graphic below.

More about the Number 1556:

1556 is the sum of two squares:
34² + 20² = 1556.

1556 is the hypotenuse of a Pythagorean triple:
756 -1360-1556, which is 4 times (189-340-389).
It can also be calculated from 34² – 20², 2(34)(20), 34² + 20².

1556 is also the difference of two squares:
390²  – 388²  = 1556.