factor pairs

What Kind of Shape is 925 In?

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925 is the 22nd Centered Square number because 22² + 21² = 925. I made this graphic to show this fact through the use of color. Look at the center of the centered square. Can you see how the single yellow square and the four small green squares in the center correspond to the same colored squares in the smaller squares? The pattern continues from the inside of the centered square to the outside.

925 is the sum of two squares these THREE ways:

 

  • 22² + 21² = 925
  • 27² + 14² = 925
  • 30² + 5² = 925

925 is the hypotenuse of SEVEN Pythagorean triples:

  • 43-924-925
  • 259-888-925
  • 285-880-925
  • 300-875-925
  • 520-765-925
  • 533-756-925
  • 555-740-925

925 is the 25th pentagonal number because 3((25²) – 25)/2 = 925. The shape in the graphic below of 925 tiny squares may look more like a house, but it is still very much a pentagon.

925 looks interesting in a few other bases:

4141 in BASE 6
1K1 BASE 22 (K is 20 in base 10)
151 BASE 28
PP in BASE 36 (P is 25 in base 10)

  • 925 is a composite number.
  • Prime factorization: 925 = 5 × 5 × 37, which can be written 925 = 2² × 37
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 925 has exactly 6 factors.
  • Factors of 925: 1, 5, 25, 37, 185, 925
  • Factor pairs: 925 = 1 × 925, 5 × 185, or 25 × 37
  • Taking the factor pair with the largest square number factor, we get √925 = (√25)(√37) = 5√37 ≈ 30.41381

 

Applying Divisibility Rules to 924

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Divisibility Rules and 924:

Let’s apply some basic divisibility rules to find some of the factors of 924:

  1. Like every other counting number, 924 is divisible by 1.
  2. Since 924 is even, it is divisible by 2.
  3. 9 + 2 + 4 = 15, a number divisible by 3, so 924 is divisible by 3.
  4. Its last two digits, 24, is divisible by 4, so 924 is divisible by 4.
  5. Its last digit isn’t 0 or 5, so 924 is NOT divisible by 5.
  6. 924 is even and divisible by 3, so it is also divisible by 6.
  7. Since 92-2(4) = 84, a number divisible by 7, we know that 924 is also divisible by 7.
  8. Since its last two digits are divisible by 8, and the third to the last digit, 9, is odd, 924 is NOT divisible 8.
  9. 9 + 2 + 4 = 15, a number not divisible by 9, so 924 is NOT divisible by 9.
  10. The last digit is not 0, so 924 is NOT divisible by 10.
  11. 9 – 2 + 4 = 11, so 924 is divisible by 11.

Thus, 1, 2, 3, 4, 6, 7, and 11 are all factors of 924.

Factor Cake for 924:

You can see its prime factors easily on the outside of its festive prime factor cake:

Factors of 924:

  • 924 is a composite number.
  • Prime factorization: 924 = 2 x 2 x 3 x 7 x 11, which can be written 924 = 2² x 3 x 7 x 11
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 x 2 = 24. Therefore 924 has exactly 24 factors.
  • Factors of 924: 1, 2, 3, 4, 6, 7, 11, 12, 14, 21, 22, 28, 33, 42, 44, 66, 77, 84, 132, 154, 231, 308, 462, 924
  • Factor pairs: 924 = 1 x 924, 2 x 462, 3 x 308, 4 x 231, 6 x 154, 7 x 132, 11 x 84, 12 x 77, 14 x 66, 21 x 44, 22 x 42, or 28 x 33,
  • Taking the factor pair with the largest square number factor, we get √924 = (√4)(√231) = 2√231 ≈ 30.3973683.

Sum Difference Puzzle:

924 has twelve factor pairs. One of the factor pairs adds up to 65, and a different one subtracts to 65. If you can identify those factor pairs, then you can solve this puzzle!

More about the Number 924:

You may have seen one of its many possible factor trees contained in the first frame of this factor tree for 852,852 from my previous post:

924 is in the very center of the 12th row of Pascal’s triangle because 12!/(6!6!) = 924.

924 is the sum of consecutive prime numbers: 461 + 463 = 924

I like 924 written in some other bases:
770 BASE 11
336 BASE 17
220 BASE 21
SS BASE 32, S is 28
S0 BASE 33

924 has several sets of consecutive factors. Besides being divisible by the 1st, 2nd, and 3rd triangular numbers (1, 3, and 6), those consecutive factors mean the following:

  • 924 is divisible by the 6th triangular number, 21, which is 6(7)/2.
  • 924 is divisible by the 11th triangular number, 66, which is 11(12)/2.
  • 924 is divisible by the 21st triangular number, 231, which is 21(22)/2.

923 Grave Marker

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To me, graveyards are beautiful places where the dearly departed are laid to rest. Find A Grave and Billiongraves are two genealogical sources that assist individuals in finding gravesites. When my son and I visited graveyards in Hungary and Slovakia a few years ago, we saw many wood and stone grave markers that had been eroded by weather. Some were almost impossible to read. We also suspect some people were too poor when they died to get a headstone of any type. We were very excited when we saw any readable grave markers with our family surnames.

Recently on twitter, I saw these paintings of gothic graveyards by M J Forster. I knew immediately I wanted to include them in this post. The paintings are quite stunning.

Finding departed ancestors can sometimes be difficult, but very rewarding. Finding the factors in today’s puzzle will be very easy:

Print the puzzles or type the solution on this excel file: 12 factors 923-931

Here’s a fun fact about the number 923:

OEIS.org informs us that 923(923 + 1) = 852,852. Below are two of the MANY possible factor trees for 852,852. The first one includes factor trees for 923 and 924, the second one shows why their product uses digits that repeat itself in order.

  • 923 is a composite number.
  • Prime factorization: 923 = 13 × 71
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 923 has exactly 4 factors.
  • Factors of 923: 1, 13, 71, 923
  • Factor pairs: 923 = 1 × 923 or 13 × 71
  • 923 has no square factors that allow its square root to be simplified. √923 ≈ 30.380915

922 Boo!

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Have you ever cut holes in a sheet, put it over your head, and jumped out in front of people as you hollered, “Boo!”? Today’s puzzle is made to look like a ghost. It’s a level 6, but don’t let that spook you! Attack the puzzle using logic, and after you solve it, you can claim to be a ghost-buster!

Print the puzzles or type the solution on this excel file: 10-factors-914-922

I think my ghost is cute, maybe not as cute as any that might knock on your door on Halloween, but still quite cute.

When 922 floats around in a different base, you may think you’re seeing an apparition:

922 becomes 1234 in BASE 9 because 1(9³) + 2(9²) + 3(9¹) + 4(9º) = 922.
922 becomes palindrome 262 in BASE 20

922 is also the sum of the 18 prime numbers from 17 to 89.

922 = 29² + 9², so 922 is the hypotenuse of a Pythagorean triple:
522-760-922, which is the same as 2(29)(9), 29² – 9², 29² + 9².
That Pythagorean triple is also 2 times (261-380-461).

  • 922 is a composite number.
  • Prime factorization: 922 = 2 × 461
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 922 has exactly 4 factors.
  • Factors of 922: 1, 2, 461, 922
  • Factor pairs: 922 = 1 × 922 or 2 × 461
  • 922 has no square factors that allow its square root to be simplified. √922 ≈ 30.3644529

 

 

921 Is This Bug Cute or Creepy?

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Some bugs make creepy Halloween decorations. Other bugs, like ladybugs, might make a very cute costume.

Today’s puzzle looks like a bug, but there is no reason to run and hide from this one. Yes, it’s a level 5, so some parts of it may be tricky.

This is what you need to do to solve it: stay calm; don’t guess and check. Figure out where to put each number from 1 to 10 in both the top row and the first column so that the clues make the puzzle work like a multiplication table. Don’t write a number down unless you are absolutely sure it belongs where you’re putting it. Use logic, step by step, and this puzzle will be a treat.

Print the puzzles or type the solution on this excel file: 10-factors-914-922

When you put on a Halloween costume, you may look completely different.

When a number is written in a different base, it may look completely different. For example,
921 looks like repdigit 333 in BASE 17 because 3(17²) + 3(17¹) + 3(17º) = 3(289 + 17 + 1) = 3(307) = 921
(307 is 111 in BASE 17)

921 looks like palindrome 1H1 in BASE 23 (H is 17 base 10). As you might suspect, 1(23²) + 17(23¹) + 1(23º) = 529 + 391 + 1 = 921

When it’s not written in a different base, 921 looks pretty familiar. You can tell quite quickly that it is divisible by 3:

  • 921 is a composite number.
  • Prime factorization: 921 = 3 × 307
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 921 has exactly 4 factors.
  • Factors of 921: 1, 3, 307, 921
  • Factor pairs: 921 = 1 × 921 or 3 × 307
  • 921 has no square factors that allow its square root to be simplified. √921 ≈ 30.34798181

920 Witches’ Cauldron

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“Double, double toil and trouble;
Fire burn, and caldron bubble.”

What besides “eye of newt” goes in witches’ cauldrons? The list includes some horrifying ingredients that you can read here from one scene from Shakeaspeare’s play, MacBeth.

Instead of putting “Eye of newt, and toe of frog, Wool of bat, and tongue of dog” and so forth in today’s Halloween cauldron puzzle, I just put a bunch of asterisks.

Print the puzzles or type the solution on this excel file: 10-factors-914-922

“Double, double toil and trouble;
Fire burn, and caldron bubble.”

Double 115 is 230.

Double 230 is 460.

Double 460 is 920, today’s post number.

920 is the hypotenuse of a Pythagorean triple:
552-736-920 which is (3-4-5) times 184.

920 is palindrome 767 in BASE 11 because 7(121) + 6(11) + 7(1) = 920

  • 920 is a composite number.
  • Prime factorization: 920 = 2 × 2 × 2 × 5 × 23, which can be written 920 = 2³ × 5 × 23
  • 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 920 has exactly 16 factors.
  • Factors of 920: 1, 2, 4, 5, 8, 10, 20, 23, 40, 46, 92, 115, 184, 230, 460, 920
  • Factor pairs: 920 = 1 × 920, 2 × 460, 4 × 230, 5 × 184, 8 × 115, 10 × 92, 20 × 46, or 23 × 40
  • Taking the factor pair with the largest square number factor, we get √920 = (√4)(√230) = 2√230 ≈ 20.331501776.

 

918 Grim Reaper’s Scythe

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Sometime on Halloween you are likely to see the Grim Reaper carrying a scythe. Together they look pretty scary. This puzzle isn’t that bad though. You should give it a try.

Print the puzzles or type the solution on this excel file: 10-factors-914-922

Scythe, now that is a good word to try when playing hangman. ☺

Let me tell you about the number 918:

It is the sum of consecutive prime numbers: 457 + 461 = 918

It is the hypotenuse of a Pythagorean triple:
432-810-918, which is (8-15-17) times 54

918 looks interesting in a few other bases:

  • 646 in BASE 12, because 9(144) + 4(12) + 6(1) = 918
  • 330 in BASE 17, because 3(289) + 3(17) + 0(1) = 3(289 + 17) = 3(306) = 918
  • 198 in BASE 26, which is the digits of 918 in a different order. Note that 1(26²) + 9(26) + 8(1) = 918
  • RR in BASE 33, (R is 27 in base 10), because 27(33) + 27(1) = 27(33 + 1) = 27(34) = 918
  • R0 in BASE 34, because 27(34) = 918

918 has consecutive numbers, 17 and 18, as two of its factors. That means 918 is a multiple of the 17th triangular number, 153.

  • 918 is a composite number.
  • Prime factorization: 918 = 2 × 3 × 3 × 3 × 17, which can be written 918 = 2 × 3³ × 17
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 × 4 × 2 = 16. Therefore 918 has exactly 16 factors.
  • Factors of 918: 1, 2, 3, 6, 9, 17, 18, 27, 34, 51, 54, 102, 153, 306, 459, 918
  • Factor pairs: 918 = 1 × 918, 2 × 459, 3 × 306, 6 × 153, 9 × 102, 17 × 54, 18 × 51, or 27 × 34
  • Taking the factor pair with the largest square number factor, we get √918 = (√9)(√102) = 3√102 ≈ 30.29851

917 How to Solve for X with Candy

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How many m&m’s are there in one fun size Halloween pack of m&m’s?

I don’t know. You don’t know. Nobody knows. That’s why these little fun size packs make the perfect UNKNOWN. For this activity, there may even be a negative number of m&m’s in a pack because I’m only using blue and orange m&m’s, and I’m letting each blue m&m equal negative one and each orange m&m equal positive one.

I’m also letting the front side of the fun size package equal negative x, and the back side equal positive x. In algebra we often call our unknown x.

(The colors chosen don’t matter, as long as there are only two of them, and you are consistent with that color being positive or negative. The front of the package could just as easily be +1 and the back -1. Consistency is important. Choose the values you want to use and stick with them. You can also use ALL the m&m’s in a few single packs and have the side of the candy with the m be positive and the side without the m be negative. You can use the empty wrappers as x and -x.)

We can figure out how many m&m’s are in the pack by balancing an equation. The number of m&m’s in a pack is x. We will solve for x by using the very best algebra tiles in the world, m&m’s!

Besides fun size m&m’s (or Skittles or Reece’s Pieces) we need a paper balance for our equations:

Click Equation Balance for a printable pdf of the paper balance.

Now let’s solve x – 3 = 5 by using the m&m’s to find x. This is how the equation balance should look to begin:

We want to get the wrapper by itself, so what do we do? To keep the equation balanced, we add three (positive) orange m&m’s (one at a time) to both sides of the paper balance:

Three (negative) blue m&m’s plus three (positive) orange m&m’s are equal to zero, so we can remove them.

Mmm. I just ate zero m&m’s, and they tasted so good! That leaves us with x = 8, so we have found x, and the equation has been solved! (If you have more equations to solve, you might want to wait to eat the m&m’s until you’re just about finished.)

Now let’s try finding x when the equation is a little more complicated, -2x = x + 12. This is how the balance should look at the beginning:

We want to get all the x’s on one side so we subtract x from both sides of the equation by adding a (negative) front-facing wrapper to both sides of the balance.

Since x – x = 0, we can remove the front-facing and back-facing wrappers from the right side of the equation:

We can arrange the 12 candies into 3 rows of 4.

Now we can divide both sides of the equation by 3.

All that’s left to do is change the signs of EVERYTHING on both sides of the equation:

Thus x = -4. We solved for x correctly because we kept the equation balanced every step of the way.

Now let me tell you a little bit about the number 917:

917 is the sum of five consecutive prime numbers:
173 + 179 + 181 + 191 + 193 = 917

Rearrange its digits and 917 becomes 197 in BASE 26.

  • 917 is a composite number.
  • Prime factorization: 917 = 7 × 131
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 917 has exactly 4 factors.
  • Factors of 917: 1, 7, 131, 917
  • Factor pairs: 917 = 1 × 917 or 7 × 131
  • 917 has no square factors that allow its square root to be simplified. √917 ≈ 30.282007859

916 Witch’s Hat

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Today’s puzzle looks a little like a witch’s hat. Solving it could be bewitching. It contains a lot of perfect squares, and it’s only a level 2. Give it a try! Just make sure you don’t write a number greater than ten in the top row or the first column. Use all the numbers from one to ten in both places. There is only one solution.

Print the puzzles or type the solution on this excel file: 10-factors-914-922

916 looks the same upside-down as it does right-side up, so it is a Strobogrammatic number.

It looks like someone put a spell on that number!

Not only that, in BASE 26, our 916 looks like 196. There is a whole lot of magic going on here!

30² + 4² = 916, so 916 is the hypotenuse of a Pythagorean triple:
240-884-916, which can be calculated from 2(30)(4), 30² – 4², 30² + 4².

  • 916 is a composite number.
  • Prime factorization: 916 = 2 × 2 × 229, which can be written 916 = 2² × 229
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 916 has exactly 6 factors.
  • Factors of 916: 1, 2, 4, 229, 458, 916
  • Factor pairs: 916 = 1 × 916, 2 × 458, or 4 × 229
  • Taking the factor pair with the largest square number factor, we get √916 = (√4)(√229) = 2√229 ≈ 30.2654919

 

 

915 Traditional Vampire Deterrent

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915 is the hypotenuse of four Pythagorean triples:

  • 165-900-915 which is 15 times (11-60-61)
  • 408-819-915 which is 3 times (136-273-305)
  • 621-672-915 which is 3 times (207-224-305)
  • 549-732-915 which is (3-4-5) times 183

 

Print the puzzles or type the solution on this excel file: 10-factors-914-922

Here’s a little more about the number 915:

915 is repdigit 555 in BASE 13 because 5(13²) + 5(13) + 5(1) = 5(183) = 915.

It is palindrome 393 in BASE 16 because 3(16²) + 9(16) +3(1) = 915.

And as 195 in BASE 26, it uses its base 10 digits in a different order. Note that 1(26²) + 9(26) + 5(1) = 915.

  • 915 is a composite number.
  • Prime factorization: 915 = 3 × 5 × 61
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 915 has exactly 8 factors.
  • Factors of 915: 1, 3, 5, 15, 61, 183, 305, 915
  • Factor pairs: 915 = 1 × 915, 3 × 305, 5 × 183, or 15 × 61
  • 915 has no square factors that allow its square root to be simplified. √915 ≈ 30.2489669.