Factors

957 Mystery Pentagon Puzzle

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Pentagons have been on my mind this week so here is another puzzle with a pentagon in it. This time the pentagon is small. How difficult is this Mystery Level puzzle?  That depends on if you recognize one very important piece of logic needed to solve it. If you see that logic, it’s not too bad. If you don’t, it might do you in.

Print the puzzles or type the solution in this excel file: 10-factors-951-958

957 is the hypotenuse of a Pythagorean triple:
660-693-957 which is (20-21-29) times 33

957 is repdigit TT in BASE 32 (T is 29 base 10)
because 29(32) + 29(1) = 29(32 + 1) = 29(33) = 957
957 is also T0 in BASE 33 because 29(33) + 0(1) = 957

  • 957 is a composite number.
  • Prime factorization: 957 = 3 × 11 × 29
  • 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 957 has exactly 8 factors.
  • Factors of 957: 1, 3, 11, 29, 33, 87, 319, 957
  • Factor pairs: 957 = 1 × 957, 3 × 319, 11 × 87, or 29 × 33
  • 957 has no square factors that allow its square root to be simplified. √957 ≈ 30.9354

956 A Pentagonal Puzzle for Paula Beardell Krieg

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Paula Beardell Krieg likes mathematics. She is also an expert paper folder. Lately, she has been turning pentagons into five-point stars. Here are a couple of her recent twitter posts:

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js

On Monday I wrote about a centered pentagonal number and included a graphic. Paula saw the post and tweeted:

https://platform.twitter.com/widgets.js

Paula followed through and made my graphic into a beautiful five-point star:

https://platform.twitter.com/widgets.js

There is nothing like trying to do origami for the first time to make me realize how irregular my pentagon graphic is. I would call my first attempt an epic fail. Sorry, I didn’t take any pictures.

However, before I started folding anything, I made this puzzle for Paula because she inspired me to make a puzzle with a pentagon in it. I needed the puzzle to be at least a 13 x 13 puzzle to get the large pentagon in it, but I decided to make it a 14 x 14 instead. It’s a level 5 so there will be some tricky parts, especially since most of the multiples of 7 in the puzzle are also multiples of 14. Don’t let that stop you from trying to solve it.

Print the puzzles or type the solution in this excel file: 10-factors-951-958

Anyways, after making the puzzle and making my epic fail star, I made this closer-to-regular pentagon on my computer:

 

I made a star using it. It looked pretty good so I decided to give my graphic of centered pentagonal number 951 a second try.  I cut it to make it more regular. Then I followed the directions on the video Paula recommended. My previous folds caused me some problems, but I was able to make something that looks like a star. It isn’t as good as Paula’s, especially on the back, but I’m okay with it. Here are pictures, front and back, of both stars I made (flaws and all):

Now since this is my 956 post, I will share some information about that number:

956 is a palindrome in two other bases:
4C4 BASE 14 (C is 12 base 10) because 4(14²) + 12(14¹) + 4(14⁰) = 956
2H2 BASE 18 (H is 17 base 10) because 2(18²) + 17(18¹) + 2(18⁰) = 956

  • 956 is a composite number.
  • Prime factorization: 956 = 2 × 2 × 239, which can be written 956 = 2² × 239
  • 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 956 has exactly 6 factors.
  • Factors of 956: 1, 2, 4, 239, 478, 956
  • Factor pairs: 956 = 1 × 956, 2 × 478, or 4 × 239
  • Taking the factor pair with the largest square number factor, we get √956 = (√4)(√239) = 2√239 ≈ 30.9192

 

955 and Level 4

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This level 4 puzzle has ten clues and two of them are the same number! No worries! There is still only one way to write the numbers from 1 to 10 in the first column and the top row so that this puzzle will function like a multiplication table. Can you figure out where to place those numbers?

Print the puzzles or type the solution in this excel file: 10-factors-951-958

Here’s some information about the number 955:

It is the hypotenuse of a Pythagorean triple:
573-764-955 which is (3-4-5) times 191

It is a palindrome in two other bases:
32323 in BASE 4 because 3(4⁴) + 2(4³) + 3(4²) + 2(4¹) + 3(4⁰) = 955
353 BASE 17 because 3(17²) + 5(17¹) + 3(17⁰) = 955

  • 955 is a composite number.
  • Prime factorization: 955 = 5 × 191
  • 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 955 has exactly 4 factors.
  • Factors of 955: 1, 5, 191, 955
  • Factor pairs: 955 = 1 × 955 or 5 × 191
  • 955 has no square factors that allow its square root to be simplified. √955 ≈ 30.90307

954 and Level 3

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The ten clues in this puzzle are all that is needed to solve the puzzle and then make it a complete multiplication table. Seriously, ten clues! Yes, the factors and the products will not be in their usual places, but the completed puzzle will still be a legitimate multiplication table. Can YOU make it work?

Print the puzzles or type the solution in this excel file: 10-factors-951-958

954 is the sum of ten consecutive prime numbers:
73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 = 954

27² +  15² = 954 so 954 is the hypotenuse of a Pythagorean triple:
504-810-954 which is 18 times (28-45-53)

954 is a palindrome in two consecutive bases:
676 in BASE 12 because 6(144) + 7(12) + 6(1) = 954
585 in BASE 13 because 5(169) + 8(13) + 5(1) = 954

  • 954 is a composite number.
  • Prime factorization: 954 = 2 × 3 × 3 × 53, which can be written 954 = 2 × 3² × 53
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 954 has exactly 12 factors.
  • Factors of 954: 1, 2, 3, 6, 9, 18, 53, 106, 159, 318, 477, 954
  • Factor pairs: 954 = 1 × 954, 2 × 477, 3 × 318, 6 × 159, 9 × 106, or 18 × 53,
  • Taking the factor pair with the largest square number factor, we get √954 = (√9)(√106) = 3√106 ≈ 30.88689

953 and Level 2

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This level 2 puzzle is only a tiny bit more difficult than a level 1 puzzle is. Start by finding the common factors of 4, 12, 40, and 28. The common factors are 1, 2, and 4, but 4 is the only one that works for the puzzle because we aren’t allowed to put factors like 14 or 28 in the top row. We are only allowed to write factors from 1 to 10 in the first column or the top row. Give this puzzle a try. I’m confident you can solve it!

Print the puzzles or type the solution in this excel file: 10-factors-951-958

Prime number 953 is the sum of the 21 prime numbers from 7 to 89.

28² + 13² = 953, so 953 is the hypotenuse of a Pythagorean triple:
615-728-953 calculated from 28² – 13², 2(28)(13), 28² + 13²

953 is a palindrome in base 11 and base 28:
797 in BASE 11 because 7(11²) + 9(11¹) + 7(11⁰) = 953
161 in BASE 28 because 1(28²) + 6(28¹) + 1(28⁰) = 953

953 × 19 × 3 = 54321, making 953 its biggest prime factor. Thank you OEIS.org
for that fun fact.

  • 952 is a prime number.
  • Prime factorization: 952 is prime.
  • The exponent of prime number 952 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 952 has exactly 2 factors.
  • Factors of 952: 1, 952
  • Factor pairs: 952 = 1 × 952
  • 952 has no square factors that allow its square root to be simplified. √952 ≈ 30.854497

How do we know that 952 is a prime number? If 952 were not a prime number, then it would be divisible by at least one prime number less than or equal to √952 ≈ 30.9. Since 952 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 952 is a prime number.

Here’s another way we know that 952 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 28² + 13² = 952 with 28 and 13 having no common prime factors, 952 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √952 ≈ 30.9. Since 952 is not divisible by 5, 13, 17, or 29, we know that 952 is a prime number.

 

 

 

952 and Level 1

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If you’ve never solved a Find the Factors puzzle before, this one will be perfect for you to do. It only has nine clues, but that is enough to find all the factors and fill in the entire multiplication table. You will feel quite smart when you’re done, too.

Print the puzzles or type the solution in this excel file: 10-factors-951-958

This is my 952nd post so I will mention a few facts about that number.

OEIS.org informs us that 93 + 53 + 23 + 9 × 5 × 2 = 952.

952 is the hypotenuse of a Pythagorean triple:
448-840-952 which is (8-15-17) times 56

952 looks interesting in some other bases:
4224 in BASE 6 because 4(6³) + 2(6²) + 2(6¹) + 4(6⁰) = 952
2C2 in BASE 19 (C is 12 Base 10) because 2(19²) + 12(19¹) +2(19⁰) = 952
SS BASE 33 (S is 28) because 28(33¹) + 28(33⁰) = 28(33 + 1) = 28(34) = 952
S0 BASE 34 because 28(34¹) + 0(34⁰) = 28(34) = 952

  • 952 is a composite number.
  • Prime factorization: 952 = 2 × 2 × 2 × 7 × 17, which can be written 952 = 2³ × 7 × 17
  • 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 952 has exactly 16 factors.
  • Factors of 952: 1, 2, 4, 7, 8, 14, 17, 28, 34, 56, 68, 119, 136, 238, 476, 952
  • Factor pairs: 952 = 1 × 952, 2 × 476, 4 × 238, 7 × 136, 8 × 119, 14 × 68, 17 × 56, or 28 × 34
  • Taking the factor pair with the largest square number factor, we get √952 = (√4)(√238) = 2√238 ≈ 29.854497

951 is the 20th Centered Pentagonal Number

/ ivasallay / 3 Comments

Since 951 is the 20th centered pentagonal number, I decided to make the following graphic with 20 concentric pentagons. I’ve outlined the pentagons in the center to make them clearer. The graphic also shows that 951 is one more than five times the 19th triangular number.

951 is also the hypotenuse of a Pythagorean triple:
225-924-951 which is 3 times (75-308-317)

As numbers get bigger, palindromes in base 2 get rarer, but 951 is one of them:
1110110111 in BASE 2 because 1(2⁹) + 1(2⁸) + 1(2⁷) + 0(2⁶) + 1(2⁵) + 1(2⁴) + 0(2³) + 1(2²) + 1(2¹) +1(2⁰) = 951
It is also 1D1 in BASE 25 (D is 14 base 10) because 1(25²) + 13(25¹) + 1(25⁰) = 951

  • 951 is a composite number.
  • Prime factorization: 951 = 3 × 317
  • 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 951 has exactly 4 factors.
  • Factors of 951: 1, 3, 317, 951
  • Factor pairs: 951 = 1 × 951 or 3 × 317
  • 951 has no square factors that allow its square root to be simplified. √951 ≈ 30.83828789

950 and Level 6

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Today’s puzzle is a level 6, so it isn’t for beginners. It is trickier than the easier levels, but don’t let that stop you from giving it a try. If you use guess and check, you will likely become frustrated, but you can solve this puzzle IF you use logic before you write each factor.

Print the puzzles or type the solution in this excel file: 12 factors 942-950

950 is the hypotenuse of two Pythagorean triples:
266-912-950 which is (7-24-25) times 38
570-760-950 which is (3-4-5) times 190

950 is 2525 in BASE 7 because 2(7³) + 5(7²) + 2(7¹) + 5(7⁰) = 950

  • 950 is a composite number.
  • Prime factorization: 950 = 2 × 5 × 5 × 19, which can be written 950 = 2 × 5² × 19
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 950 has exactly 12 factors.
  • Factors of 950: 1, 2, 5, 10, 19, 25, 38, 50, 95, 190, 475, 950
  • Factor pairs: 950 = 1 × 950, 2 × 475, 5 × 190, 10 × 95, 19 × 50, or 25 × 38,
  • Taking the factor pair with the largest square number factor, we get √950 = (√25)(√38) = 5√38 ≈ 30.82207

 

949 Mystery Level

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The difficulty level of a mystery level puzzle is a mystery. You won’t know how hard or how easy it is until you give it a try. You can solve it by using logic and your knowledge of the multiplication table. Can you figure it out?

Print the puzzles or type the solution in this excel file: 12 factors 942-950

Now here is a little about the number 949:

948 and 949 are a Ruth-Aaron pair.

949 is not only a palindrome in base 10, but it is also
434 in BASE 15 because 4(15²) + 3(15¹) + 4(15⁰) = 949

25² + 18² = 949 and 30² + 7² = 949 so 949 is the hypotenuse of four Pythagorean triples:

301-900-949 calculated from 25² – 18², 2(25)(18), 25² + 18²
365-876-949 which is (5-12-13) times 73
420-851-949 calculated from 2(30)(7), 30² – 7², 30² + 7²
624-715-949 which is 13 times (48-55-73)

  • 949 is a composite number.
  • Prime factorization: 949 = 13 × 73
  • 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 949 has exactly 4 factors.
  • Factors of 949: 1, 13, 73, 949
  • Factor pairs: 949 = 1 × 949 or 13 × 73
  • 949 has no square factors that allow its square root to be simplified. √949 ≈ 30.80584

948 and Level 5

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All you need is a little bit of logic and the multiplication facts in a standard 12 x 12 multiplication table to solve this puzzle. This puzzle is not just for kids. It can be challenging even for adults. Go ahead, give it a try. You’ll soon see that it’s tougher than it looks, but stick with it, you can conquer it!

Print the puzzles or type the solution in this excel file: 12 factors 942-950

Here’s something cool about the number 948 that I learned from OEIS.org:

948 and 949 make a Ruth-Aaron pair because they are consecutive numbers and the sum of 948’s prime factors equals the sum of 949’s prime factors:

  • 948 is a composite number.
  • Prime factorization: 948 = 2 × 2 × 3 × 79, which can be written 948 = 2² × 3 × 79
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 948 has exactly 12 factors.
  • Factors of 948: 1, 2, 3, 4, 6, 12, 79, 158, 237, 316, 474, 948
  • Factor pairs: 948 = 1 × 948, 2 × 474, 3 × 316, 4 × 237, 6 × 158, or 12 × 79,
  • Taking the factor pair with the largest square number factor, we get √948 = (√4)(√237) = 2√237 ≈ 30.7896