# Facts, Factors, and a Countdown to 2023

### Countdown to 2023:

The last 10 seconds of the year, we like to countdown from 10 to the new year. I like a mathematical way of counting down so I try to make an equation with the numbers from 10 to 1 that equals the coming year. This year I could have based my countdown  on last year’s countdown and said
(10-9+8×7×6)(5-4)(3)(2)+1 = 2023,
but this is a blog about factoring so I want a countdown that takes you to the prime factors of 2023 first. Here’s my countdown: (Note: Even though I used 1 as a factor twice in the countdown, I am very much aware that 1 is not a prime factor of any number.)

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Here’s the final shot from the Countdown:

### A Puzzle for 2023:

2023 has something in common with 2022. When either number and their reverses are squared, something interesting happens…it’s almost like looking in a mirror!

Only 50 numbers less than 10000 can make a similar claim to fame:

### Factors of 2023:

• 2023 is a composite number.
• Prime factorization: 2023 = 7 × 17 × 17, which can be written 2023 = 7 × 17².
• 2023 has at least one exponent greater than 1 in its prime factorization so √2023 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √2023 = (√289)(√7) = 17√7.
• The exponents in the prime factorization are 1 and 2. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1) = 2 × 3 = 6. Therefore 2023 has exactly 6 factors.
• The factors of 2023 are outlined with their factor pair partners in the graphic below.

### More About the Number 2023:

What do 2023 tiny squares look like?

2023 is the sum of consecutive numbers in five different ways:

And it is the sum of consecutive odd numbers in two ways:

2023 is a palindrome in base 16 because
7(16²) + 14(16) + 7(1) = 2023.

This tweet demonstrates that the prime factors of 2023 have a relationship with the digits of 2023.

I saw this tweet on Twitter and immediately recognized something else cool about 2023.

That might seem like a lot of mathematical mumble jumble, but with a little bit of explanation, it can be understood. And even though I made the problem look scarier because I substituted 2+0+2+3 for 7, some older elementary students who already understand powers, factorials, and/or remainders, will get it. I’m confident you can too.

You could also give the following list of facts to older elementary students and ask them to use it to find the remainder when they divide 823,543 by 5040.

1×5040 =    5040
2×5040 = 10080
3×5040 = 15120
4×5040 = 20160
5×5040 = 25200
6×5040 = 30240
7×5040 = 35280
8×5040 = 40320
9×5040 = 45360

Because its factors, 17 and 289, are hypotenuses of Pythagorean triples, 2023 is also the hypotenuse of some Pythagorean triples:
952-1785-2023 which is 119(8-15-17) , and
1127-1680-2023 which is 7(161-240-289).

Ureczky József also pointed out in the comments of this post, that 2023 is the short leg in SIX Pythagorean triples, and thus
2023² = 2046265² – 2046264²
2023²= 292327² – 292320²
2023² = 120377² – 120360²
2023² = 17255² – 17136²
2023² = 41785² – 41736²
2023² = 7225² – 6936²

One of those triples is a primitive triple. Can you determine which one?

Ureczky József shared one more amazing fact in the comments that I’m replicating here:

2023=9³+8³+7³+6³+5³+4³+3³+2³-1³

### More Mathematical Tweets About 2023:

These tweets are more or less in the order I saw them, not in order of mathematical difficulty. I will add more as I see them.

v

# 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:

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### It’s Almost Like 2021 Is Looking in a Mirror!

Squaring means multiplying a number by itself. Something unusual happens when you square 2021 and you also square its reverse, 1202. Complete the two multiplication problems below to see for yourself!

There are only 50 numbers less than 10000 with this same property. They are all listed in the table below:

For each pair of numbers in the table, the smaller number was 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.

### Math Facts about the Number 2021:

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Here’s a Magic Square showing 2021:

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.

### 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.

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

# 1009, a Prime Factor of the Year 2018

Let’s begin with a mathematical equation you can use to countdown the final seconds of 2017 to welcome in the New Year, 2018.

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*********

Okay, that equation had a lot of parentheses which made us multiply by 1 two different times. (Boring.) I designed it the way it is because I wanted to take advantage of the fact that 1009 × 2 = 2018.

This second equation created by Edmark M. Law needs no parentheses and is much more beautiful:

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**********

Sometimes you might need a still picture instead of a gif.

This post will include lots of facts about 2018.

2018¹⁰⁻⁹⁻⁸⁺⁷⁺⁶⁻⁵⁻⁴⁺³⁺²⁻¹ = 2018

2018 can be written as the sum of four consecutive numbers:
503 + 504 + 505 + 506 = 2018

It can also be written as the sum of two consecutive even numbers:
1008 + 1010 = 2018

2018 will be an amazing year in many different areas:

That graphic is based on 43² +  13² = 2018.

This next one is based on the fact that 2(1009) = 2(28² +  15²) = 2018.

2018 is also the sum of three squares four different ways:

36² +  19² +  19² = 2018
35² +  28² +  3² = 2018
35² +  27² +  8² = 2018
33² +  23² +  20² = 2018

This next area problem is based on one of the sums of three squares listed above. Can you tell which one?

2018 is in exactly two Pythagorean triple triangles:
1118-1680-2018 and 2018-1018080-1018082.

In the triangle illustrated above, 2018 is about 500 times smaller than either of the other two sides. Yep, that graphic was definitely not drawn to scale.

By contrast, in this next triangle, hypotenuse 2018 is not even twice as big as either of the legs.

How did I find that triangle?
1118-1680-2018 can be calculated from 2(43)(13), 43² – 13², 43² + 13²
It is also 2 times (559-840-1009). That primitive triple can be calculated from
28² – 15², 2(28)(15), 28² + 15²

Is there any other relationship between 43² + 13² and 2(28² + 15²)? Yes.
28 + 15 = 43 and 28 – 15 = 13.

How did I find the triple with two sides in the millions (2018-1018080-1018082)?
2018 ÷ 2 = 1009 and 1009² ± 1 are the values of the other leg and the hypotenuse.

I like the way 2018 looks in these other bases:
It’s 8E8 in BASE 15 (E is 14 base 10) because 8(15²) + 14(15) + 8(1) = 2018,
2G2 in BASE 28 (16 is G base 10) because 2(28²) + 16(28) + 2(1) = 2018, and
2202202 in BASE 3 because 2(3⁶ + 3⁵ + 3³ + 3² +3⁰) = 2(1009) = 2018

This is how we can write 2018 as the sum of powers of 2:
2¹⁰ + 2⁹ + 2⁸ + 2⁷ + 2⁶ + 2⁵ + 2¹ = 2018

Finally, I give you my predictions of the factors we will see in 2018. You can be confident that these predictions will be 100% correct.
The positive factors for the year 2018 will be (drum roll) 1, 2, 1009, and 2018.
Sorry to say, but there will also be four negative factors of 2018: -1, -2, -1009, and -2018.

I also know that 2018 will have some complex factors because 43² +  13² = 2018.

Here is a graphic showing 2018’s factor pairs:

Related Articles:

1. Edmark M. Law’s post titled Happy New Year 2018! (And Mathematical Facts about 2018) has many more mathematical curiosities about 2018.
2. Mathwithbaddrawings.com humorously shares some upcoming mathematical dates and other facts about 2018 in Things to Know About the Year 2018.  At least one of those facts makes the number 2018 quite unique.
3. 2018: Top Ten Facts about the New Year has a little bit of mathematics in it.

Since this is my 1009 post, I’ll tell you a few things about that number:

1009 is half of 2018.

1009 is the smallest four-digit prime number.

28² + 15² = 1009 so we get this Pythagorean triple:
559-840-1009

1009 is a palindrome or otherwise looks interesting in some other bases:
It’s 838 in BASE 11 because 8(121) + 3(11) + 8(1) = 1009,
474 in BASE 15 because 4(15²) + 7(15) + 4(1) = 1009,
321 in BASE 18 because 3(18²) + 2(18) + 1(1) = 1009,
2F2 in BASE 19 (F is 15 base 10) because 2(19²) + 15(19) + 2(1) = 1009,
1I1 in BASE 24 (I is 18 base 10) because 1(24²) + 18(24) + 1(1) = 1009, and
181 in BASE 28 because 1(28²) + 8(28) + 1(1) = 1009

• 1009 is a prime number.
• Prime factorization: 1009 is prime.
• The exponent of prime number 1009 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1009 has exactly 2 factors.
• Factors of 1009: 1, 1009
• Factor pairs: 1009 = 1 × 1009
• 1009 has no square factors that allow its square root to be simplified. √1009 ≈ 31.76476

How do we know that 1009 is a prime number? If 1009 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1009 ≈ 31.8. Since 1009 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1009 is a prime number.

Here’s another way we know that 1009 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 28² + 15² = 1009 with 28 and 15 having no common prime factors, 1009 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1009 ≈ 31.8. Since 1009 is not divisible by 5, 13, 17, or 29, we know that 1009 is a prime number.

Now I’ll include posts from twitter that celebrate 2018 mathematically. Some are easier to understand than others: