The sum of the proper divisors of a number determines if the number is abundant, deficient, or perfect. If the sum is greater than the number, the number is abundant. If the sum is less than the number, the number is deficient. If the sum is equal to the number, the number is perfect.

What is a proper divisor? All the factors of a number except itself. Proper divisors are ALMOST the same thing as proper factors. (The number 1 is always a proper divisor, but NEVER a proper factor.)

The first 25 abundant numbers are 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, and 108. Notice that all those numbers are even.

OEIS informs us that 945 is the 232nd abundant number. The first 231 abundant numbers are all even numbers.

Wow, 945 is the smallest ODD abundant number. OEIS also lists the first 31 odd abundant numbers. Every one of the first 31 is divisible by 3 and ends with a 5, but if you scroll down the page you’ll see some that aren’t divisible by 3 or aren’t divisible by 5.

Since 1 × 3 × 5 × 7 × 9 = 945 is the smallest number on the list, you may be wondering about some other numbers:
1 × 3 × 5 × 7 × 9 × 11 = 10,395 made the list.
1 × 3 × 5 × 7 × 9 × 11 × 13 = 135,135 which is too big to be one of the first 31 odd abundant numbers. I was curious if it is also an abundant number, so I found its proper divisors and added them up:

945 is also the hypotenuse of a Pythagorean triple:
567-756-945 which is (3-4-5) times 189

945 looks interesting in a few other bases:
1661 in BASE 8 because 1(8³) + 6(8²) + 6(8¹) + 1(8⁰) = 945
RR in BASE 34 (R is 27 base 10), because 27(34¹) + 27(34⁰) = 27(35) = 945
R0 in BASE 35 because 27(35) + 0(1) = 945

945 is a composite number.

Prime factorization: 945 = 3 × 3 × 3 × 5 × 7, which can be written 945 = 3³ × 5 × 7

The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 945 has exactly 16 factors.

840 has more factors than any previous number. Those factors will help us write 840 as the sum of consecutive counting numbers, consecutive even numbers, and consecutive odd numbers. What are the factors of 840? Here are a couple of the many possible factor trees for 840:

The red leaves on the tree are prime numbers. Gathering the six red leaves from either factor tree above gives us 840’s prime factorization: 840 = 2³ × 3 × 5 × 7. Now 840 is not the smallest number to have six red leaves. In fact, there are smaller numbers with as many as nine leaves, but 840’s six innocent-looking red leaves will turn into a whopping 32 factors!

Factors of 840:

840 is a composite number.

Prime factorization: 840 = 2 × 2 × 2 × 3 × 5 × 7, which can be written 840 = 2³ × 3 × 5 × 7.

The exponents in the prime factorization are 3, 1, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) )(1 + 1) = 4 × 2 × 2 × 2 = 32. Therefore 840 has exactly 32 factors.

Taking the factor pair with the largest square number factor, we get √840 = (√4)(√210) = 2√210 ≈ 28.98275.

840 is also the smallest number that can be evenly divided by the first eight counting numbers!

Sum-Difference Puzzles:

210 has eight factor pairs. One of those factor pairs adds up to 29, and another one subtracts to 29. Another factor pair adds up to 37, and a different one subtracts to 37. Thus, 210 produces two different Sum-Difference Puzzles shown in the first two graphics below!

840 is a multiple of 210 and has sixteen factor pairs! One of the factor pairs adds up to 58 and another to 74. A different factor pair subtracts to 58, and still a different one subtracts to 74. If you can identify those factor pairs, then you can solve the two puzzles that are next to the 210 puzzles below!

840 has yet another factor pair that adds up to 113 and a different one that subtracts to 113. If you can find those factor pairs, then you can solve this next primitive puzzle:

If you need help with any of those 840 Sum-Difference puzzles, the chart below shows the sums and differences of all of 840’s factor pairs. See which sums also appear in the difference column.

How to find consecutive counting numbers that add up to 840:

840 is more than the 40th triangular number (820) and less than the 41st triangular number (861). We can also arrive at the number 40 by using √(1 + 840×2) – 1= 40, no rounding necessary. 840 has six odd factors that are not more than 40, namely 1, 3, 5, 7, 15, 21 and 35.

I’ll describe the ways we can write 840 as the sum of consecutive numbers. Can you write out the sums? I’ve done one of them for you:

using 3 numbers with 280 as the middle number,

using 5 numbers with 168 as the middle number,

using 7 numbers with 120 as the middle number; 117 + 118 + 119 + 120 + 121 + 122 + 123 = 840

using 15 numbers with 56 as the middle number,

using 21 numbers with 40 as the middle number

using 35 numbers with 24 as the middle number.

Notice each of those ways has a factor pair of 840 in the description.

The largest power of 2 that is a factor of 840 is 8, which doubled becomes 16. Which of 840’s odd factors multiplied by 16 are not more than 40? 1 × 16 = 16, and 3 × 16 = 48. Oops, that’s too much. The rest of its odd factors times 16 will be too much as well. Nevertheless, we can write 840 as the sum of 16 counting numbers. 840÷16 = 52.5 so 52 and 53 will be the 8th and 9th numbers in the sum.

How do we find consecutive EVEN numbers that add up to an even number?

Only even numbers can be the sum of consecutive even numbers. Let’s use 840 as an example again. 840÷2 = 420.

First, we will find all the ways to write 420 as the sum of consecutive numbers. Then we will simply double the middle number and surround it with the appropriate number of even numbers to get a sum of even numbers that add up to 840:

√(1 + 420×2) – 1 = 28, no rounding necessary, so we will make a list of the odd factors that are not more than 28. They are 1, 3, 5, 7, 15, and 21. We also note that the largest power of 2 that is a factor of 420 is 4. Doubling 4, we get 8. Which of 420’s odd factors multiplied by 8 are not more than 28? 1 × 8 = 8, and 3 × 8 = 24. All of the rest will be too much.

I’ll describe the ways we can write 840 as the sum of consecutive even numbers. You can see 420’s factor pairs and 840’s factor pairs in the descriptions. Can you write out the sums? I’ve done one of them for you:

using 3 even numbers with 2 × 140 = 280 as the middle number

using 5 even numbers with 2 × 84 = 168 as the middle number

using 7 even numbers with 2 × 60 = 120 as the middle number

using 15 even numbers with 2 × 28 = 56 as the middle number

We can also write 840 as the sum of an even amount of consecutive even numbers.

using 8 even numbers: 2 times (49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 = 420) becomes 98 + 100 + 102 + 104 + 106 + 108 + 110 + 112 = 840. Notice that 840÷8 = 105, the odd number that is between the two numbers in the exact middle of the sum.

Likewise, using 24 even numbers: 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 + 30 + 32 + 34 + 36 + 38 + 40 + 42 + 44 + 46 + 48 + 50 + 52 + 54 + 56 + 58 = 840. Notice that 840÷ 24 = 35, the odd number that is between the two numbers in the exact middle of the sum.

How to find consecutive ODD numbers that add up to an even number like 841:

Let me first point out that every square number, n², is the sum of the first n numbers.

For example: 100 = 10², and 100 is also the sum of the first 10 odd numbers as illustrated below:

A similar statement is true for every other square number:

Yes, every square number, n², is the sum of the first n odd numbers.

To write a non-square even number as the sum of consecutive odd numbers, the first thing we must do is determine if the number can be written as the difference of two squares. If an even number has a factor pair, b × a, in which b > a, and BOTH a and b are even, then that even number can be expressed as the difference of two squares by using [(b + a)/2]² – [(b – a)/2]².

Now as long as an even number can be expressed as the difference of two squares, B² –A², then that number can be written as the sum of all the odd numbers from 2A+1 to 2B-1.

840 is an even number with eight factor pairs in which both numbers are even. Let’s use those even factor pairs to find the ways to write 840 as the sum of consecutive ODD numbers:

420 × 2 = 840 means 211²–209² = 840, and that means the sum of the 2 consecutive odd numbers from 419 to 421 = 840

210 × 4 = 840 means 107²–103² = 840, and that means the sum of the 4 consecutive odd numbers from 207 to 213 = 840

140 × 6 = 840 means 73² –67² = 840, and that means the sum of the 6 consecutive odd numbers from 135 to 145 = 840

84 × 10 = 840 means 47²–37² = 840, and that means the sum of the 10 consecutive odd numbers from 75 to 93 = 840

70 × 12 = 840 means 41²–29² = 840, and that means the sum of the 12 consecutive odd numbers from 59 to 81 = 840

60 × 14 = 840 means 37²–23² = 840, and that means the sum of the 14 consecutive odd numbers from 47 to 73 = 840

42 × 20 = 840 means 31²–11² = 840, and that means the sum of the 20 consecutive odd numbers from 23 to 61 = 840

30 × 28 = 840 means 29²–1² = 840, and that means the sum of the 28 consecutive odd numbers from 3 to 29 = 840

Thus, 840 with its record setting 32 factors, can be written as the sum of 7 consecutive numbers, 7 consecutive even numbers, and 8 consecutive odd numbers!

More about the Number 840:

Incidentally, being able to write 840 as the difference of two squares, eight different ways also makes 840 a leg in at least eight different Pythagorean triples. Those Pythagorean triples can be calculated using the numbers from the difference of two squares. For example, 682-840-1082 can be calculated from 2(31)(11), 31²–11², 31² + 11².

840 was the leg for those eight triples. It is possible that looking at 2(b)(a), where b × a = 420, will produce some more Pythagorean triples with 840 as the leg.

840 is also the hypotenuse of one Pythagorean triple, 504-672-840.

On a note totally unrelated to anything I’ve written above, 840 is a repdigit in two bases:

SS BASE 29 (S is 28 base 10) Note that 28(29) + 28(1) = 28(30) = 840

00 BASE 34 (0 is 24 base 10) Note that 24(34) + 24(1) = 24(35) = 840

840 is also the sum of twin prime numbers 419 and 421.

680 is a number made using only even digits. (There’s much more about 680 at the end of the post.)

Numbers ending in 0, 2, 4, 6, or 8 are even. Numbers ending in 1, 3, 5, 7 or 9 are odd. Those two simple concepts are not always easy for young children to understand.

Sometimes we teach rhymes to children to help them know the difference:

0, 2, 4, 6, 8; being EVEN is just great.

1, 3, 5, 7, 9; being ODD is just fine.

Still students in early grades struggle with the concepts of odd and even.

Another seemingly simple concept is what pairs of numbers add up to ten. That concept also isn’t as easy for children to understand as adults might think.

Donna Boucher is an elementary school math interventionist with many years experience. Besides many other topics, she is an expert on teaching adding and subtracting to first and second graders. Here are a couple of her tweets with links to her site:

Free ten-frame flash cards are available on her site to help students learn addition and subtraction facts. What a powerful way for students to learn! She also has Halloween/Thanksgiving ten-frames for sale at Teachers Pay Teachers.

As I read her post about how to use the ten-frame flash cards I wondered what would happen if we followed her instructions EXACTLY, but the ten-frames looked like this:

Children would still learn how to add and subtract, but would they also instinctively learn the difference between odd and even numbers?

Would they figure out for themselves that adding two even numbers or adding two odd numbers ALWAYS makes an even number? Or that adding an odd number and an even number together ALWAYS makes an odd number? Or would changing the ten-frames not make any difference at all? Will the mitten ten-frames only make a difference if the parent/teacher/tutor talks about the odd and even numbers?

I don’t know the answer to those questions, but I think the idea is worth trying. I’ve made Mitten Ten-Frames for all the numbers from 0 to 10. The “empty” frames have outlines of mittens to help children know if a left or a right mitten belongs there. The mitten ten-frames don’t have a second border to guide in cutting them out, so the flashcards might not look as good as Donna Boucher’s, but they should still work as flashcards. Follow Donna Boucher’s instructions exactly. If you use the mitten ten-frames, please add a comment to let me know whether or not they make any difference helping students learn the properties of odd and even numbers.

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Here’s more about the number 680:

1² + 3² + 5² + 7² + 9² + 11² + 13² + 15² = 680.

Because 5, 17, and 85 are some of its factors, 680 is the hypotenuse of four Pythagorean triples. Can you find the greatest common factor of each triple?

104-672-680

288-616-680

320-600-680

408-544-680

680 the 15th tetrahedral number. OEIS.org tells us that it is also the smallest tetrahedral number that can be made by adding two other tetrahedral numbers together, specifically the sum of the 10th and the 14th tetrahedral numbers equals this 15th tetrahedral number as shown below:

(10)(11)(12)/6 = 220

(14)(15)(16)/6 = 560

220 + 560 = 680

(15)(16)(17)/6 = 680

Finally, here is the factoring information for 680:

680 is a composite number.

Prime factorization: 680 = 2 x 2 x 2 x 5 x 17, which can be written 680 = (2^3) x 5 x 17

The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 680 has exactly 16 factors.

Prime factorization: 275 = 5 x 5 x 11, which can be written (5^2) x 11

The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2 = 6. Therefore 275 has 6 factors.

Factors of 275: 1, 5, 11, 25, 55, 275

Factor pairs: 275 = 1 x 275, 5 x 55, or 11 x 25

Taking the factor pair with the largest square number factor, we get √275 = (√11)(√25) = 5√11 ≈ 16.583

Are six-year-olds too young to learn about odd and even numbers?

Paula Beardell Krieg gave me permission to use the pictures of this flexible number line she designed in this post:

I recently read a post at mathfour.com that discussed the “basic” concept of odd and even numbers and children’s ability to understand the difference. The article made me very curious so I talked briefly to 45 first grade students about even and odd numbers. What did I find out?

Almost all of them had been introduced to the concept in kindergarten and knew that 1, 3, 5, 7, 9 are odd numbers while 2, 4, 6, 8, 10 are even.

A few accelerated learning students were able to explain to me that the one’s digit of a number determines if the number is even or odd,

But most of these first graders did not understand that fact because about a third of the students thought that 32 is odd!

One little girl explained to me how odd and even numbers alternate. She said, “If 99 is even, then 100 will be odd.” She remembered that concept but didn’t understand it well enough to apply it to the example she gave!

Even though odd and even numbers may be a difficult concept to learn, teach the concept and use it anyway. In fact, talk about it to preschoolers while you put on their socks, shoes, or mittens. One,_Two,_Buckle_My_Shoe.

Children learn to recite numbers in order before they learn how to count, and that helps them learn how to count and later how to add or subtract 1 from a number. I have tutored bewildered looking students who weren’t sure what to do with 8 + 1 = until I told them that 8 + 1 = means “what number comes right after 8 when you count?” Likewise, 8 – 1 = means “what number comes right before 8 when you count?” After hearing those questions, these students immediately knew the answer, and they didn’t count to find it.

Children who can quickly recite the odd numbers to 11 and the even numbers to 10 will have an easier time adding or subtracting two from a number. When they see 3 + 2 =, they can remember that 3 is odd and then ask themselves what odd number comes after 3. Likewise when they see 8 – 2, they can remember that 8 is even and recall that 6 is the even number right before 8.

The way I remember it, I was in second grade when I first was told that an even number plus an even number is even, an odd number plus an odd number is even, while an even number plus an odd number is odd. Any student learning to add or subtract would benefit from that tip.

Adding 3 to an odd number gives an even number, in fact, it’s the second even number after the original number. Adding 3 to an even number gives an odd number which is the second odd number after the original number. Subtracting 3 has the same rule, but substitute the word “before” for the word “after.”

Adding 4 to an odd number gives the second odd number after it while adding 4 to an even number gives the second even number after it. Subtracting 4 has a similar rule.

Adding 3 or 4 will mean additionally memorizing that 12 and 14 are even and 13 is odd, but that will be all a first grader needs to know about odd and even numbers. Later these two categories of numbers will be useful throughout their lives for many, many reasons.

What are some ways to help children to memorize odd and even numbers? Paula Beardell Krieg has designed the most captivating number line in the world.

The transformation can be done by a child or an adult. This number line that is made with envelopes is pretty enough to hang on a classroom wall, but it can fold up like a book, or be played with and changed so that real learning can take place. Paula Beardell Krieg shows several uses of it in her post, the-flux-capacity-of-an-artful-number-line, and promises to give directions on how to make one soon!

Try these rhymes: 0, 2, 4, 6, 8; Being EVEN is just great! 1, 3, 5, 7, 9; Being ODD is just fine!

Smartfirstgraders.com has several activities and rhymes to help students memorize the odd and even numbers.

Finally, if you clap when you say ODD, you will clap one time. 1 is an odd number.

If you clap when you say EVEN, you will clap two times, 2 is even.

And as mathfour.com pointed out in more detail then I’m showing here: ODD has 3 letters, and 3 is odd.

Also EVEN has 4 letters to help us remember that 4 is even.