Factors

981 Peppermint Sticks

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This time of year you can buy peppermint sticks that don’t just have red stripes, but they might have green ones, too. Today’s puzzle looks like a couple of peppermint sticks. It will be a sweet experience for you to solve it, so be sure to give it a try.

Print the puzzles or type the solution in this excel file: 12 factors 978-985

Here is some information about the number 981:

30² + 9² = 981 making 981 the hypotenuse of a Pythagorean triple:
540-819-981 which is 9 times (60-91-109) but can also be calculated from
2(30)(9), 30² – 9², 30² + 9²

981 is palindrome 171 in BASE 28 because 1(28²) + 7(28) + 1(1) = 981

  • 981 is a composite number.
  • Prime factorization: 981 = 3 × 3 × 109, which can be written 981 = 3² × 109
  • 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 981 has exactly 6 factors.
  • Factors of 981: 1, 3, 9, 109, 327, 981
  • Factor pairs: 981 = 1 × 981, 3 × 327, or 9 × 109
  • Taking the factor pair with the largest square number factor, we get √981 = (√9)(√109) = 3√109 ≈ 31.3209

980 Christmas Factor Trees

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This level 4 puzzle has 12 clues in it that are products of factor pairs in which both factors are numbers from 1 to 12. The clues make an evergreen tree, the symbol of everlasting life which is so fitting for Christmas. Can you find the factors for the given clues and put them in the right places?

Print the puzzles or type the solution in this excel file: 12 factors 978-985

Now I’ll tell you a little about the number 980:

It has eighteen factors and many possible factor trees. Here are just three of them:

28² + 14² = 980, so 980 is the hypotenuse of a Pythagorean triple:
588-784-980 which is (3-4-5) times 196, but can also be calculated from
28² – 14², 2(28)(14), 28² + 14²

I like the way 980 looks in some other bases:
It is 5A5 in BASE 13 (A is 10 base 10) because 5(13) + 10(13) + 5(1) = 980,
500 in BASE 14 because 5(14²) = 980,
SS in BASE 34 (S is 28 base 10) because 28(34) + 28(1) = 28(35) = 980
S0 in BASE 35 because 28(35) = 980

  • 980 is a composite number.
  • Prime factorization: 980 = 2 × 2 × 5 × 7 × 7, which can be written 980 = 2² × 5 × 7²
  • The exponents in the prime factorization are 2, 1 and 2. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(2 + 1) = 3 × 2 × 3 = 18. Therefore 980 has exactly 18 factors.
  • Factors of 980: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 49, 70, 98, 140, 196, 245, 490, 980
  • Factor pairs: 980 = 1 × 980, 2 × 490, 4 × 245, 5 × 196, 7 × 140, 10 × 98, 14 × 70, 20 × 49 or 28 × 35
  • Taking the factor pair with the largest square number factor, we get √980 = (√196)(√5) = 14√5 ≈ 31.30495.

978 A Gift-Wrapped Puzzle

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Today’s puzzle comes gift-wrapped just for you. Figuring out the solution to this puzzle is about as easy as ripping gift-wrap off a present, too. What is the common factor in both parts of the ribbon? There is only one answer to that question that will not put any numbers greater than 12 where the factors go. So put the factors of the clues in the first column and top row so that this puzzle becomes a multiplication table (but with the factors in a different order than usual).

Print the puzzles or type the solution in this excel file: 12 factors 978-985

Here are some facts about the number 978:

It is made from 3 consecutive numbers, 7-8-9, so it is divisible by 3.

OEIS.org reminds us that it is the sum of four consecutive fourth powers:
2⁴ + 3⁴ + 4⁴ + 5⁴ = 978

It is the sum of two consecutive prime numbers:
487 + 491 = 978

I like the way it looks when written in a couple of other bases:
Palindrome 696 in BASE 12 because 6(12²) + 9(12) + 6(1) = 978
369 in BASE 17 because 3(17²) + 6(17) + 9(1) = 978

  • 978 is a composite number.
  • Prime factorization: 978 = 2 × 3 × 163
  • 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 978 has exactly 8 factors.
  • Factors of 978: 1, 2, 3, 6, 163, 326, 489, 978
  • Factor pairs: 978 = 1 × 978, 2 × 489, 3 × 326, or 6 × 163
  • 978 has no square factors that allow its square root to be simplified. √978 ≈ 31.27299

 

What Kind of Shape is 976 in?

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If you look at a list of Centered Triangular Numbers, 976 will be the 26th number on the list. 976 is the sum of the 24th, 25th, and 26th triangular numbers. That means
(24×25 + 25×26 + 26×27)/2 = 976

There’s a formula to compute centered triangular numbers, and this one is found by using
(3(25²) + 3(25) + 2)/2 = 976

As far as the formula is concerned, it is the 25th centered triangular number even though it is the 26th number on the list. I’m calling it the 26th centered triangular number because counting numbers on a list is easier than using a formula.

976 is also the 16th decagonal number because 4(16²) – 3(16) = 976. I couldn’t resist illustrating that 10-sided figure.

16 × 61 is a palindromic expression that happens to equal 976.

976 is also a palindrome when written in some other bases:
It’s 1100011 in BASE 3 because 3⁶ + 3⁵ + 3¹ + 3⁰ = 976,
808 in BASE 11, because 8(11²) + 0(11¹) + 8(11⁰) = 976
1E1 in BASE 25 (E is 14 base 10) because 1(25²) + 14(25¹) + 1(25⁰) = 976

24² + 20² = 976 That makes 976 the hypotenuse of a Pythagorean triple:
176-960-976 calculated from 24² – 20², 2(24)(20), 24² + 20²

  • 976 is a composite number.
  • Prime factorization: 976 = 2 × 2 × 2 × 2 × 61, which can be written 976 = 2⁴ × 61
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 976 has exactly 10 factors.
  • Factors of 976: 1, 2, 4, 8, 16, 61, 122, 244, 488, 976
  • Factor pairs: 976 = 1 × 976, 2 × 488, 4 × 244, 8 × 122, or 16 × 61
  • Taking the factor pair with the largest square number factor, we get √976 = (√16)(√61) = 4√61 ≈ 31.240999

975 and Level 5

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This puzzle has eleven clues, some of which are designed to trick you possibly. Will you be tricked, or will you use logic to figure out where to put the factors from 1 to 10 in the first column and the top row? The finished puzzle looks like a multiplication table but with the factors out of numerical order.

Print the puzzles or type the solution in this excel file: 10-factors-968-977

Here are a few facts about the number 975:

975 is the hypotenuse of SEVEN Pythagorean triples. Some factor pairs of 975 are highlighted in red next to those triples.
108-969-975 which is 3 times (36-323-325)
240-945-975 which is 15 times (16-63-65)
273-936-975 which is (7-24-25) times 39
375-900-975 which is (5-12-13) times 75
495-840-975 which is 15 times (33-56-65)
585-780-975 which is (3-4-5) times 195
612-759-975 which is 3 times (204-253-325)

1(5⁴) + 2(5³) + 3(5²) + 4(5¹) + 5(5⁰) = 975

You might think that last fact means that 975 is 12345 in base 5, but it isn’t. The only digits used in base 5 are 0, 1, 2, 3, and 4.

Here is 975 written in some different bases:
1111001111 in BASE 2 because 2⁹ + 2⁸ + 2⁷+ 2⁶ + 2³ + 2² + 2¹ + 2⁰ = 975
33033 in BASE 4 because 3(4⁴) + 3(4³) + 0(4²) + 3(4¹) + 3(4⁰) = 3(256 + 64 + 4 + 1) = 3(325) = 975
1717 in BASE 8 because 1(8³) + 7(8²) + 1(8¹) + 7(8⁰) = 975
303 in BASE 18 because 3(18²) + 0(18¹) + 3(18⁰) = 3(324 + 1) = 3(325) = 975

  • 975 is a composite number.
  • Prime factorization: 975 = 3 × 5 × 5 × 13, which can be written 975 = 3 × 5² × 13
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 975 has exactly 12 factors.
  • Factors of 975: 1, 3, 5, 13, 15, 25, 39, 65, 75, 195, 325, 975
  • Factor pairs: 975 = 1 × 975, 3 × 325, 5 × 195, 13 × 75, 15 × 65, or 25 × 39,
  • Taking the factor pair with the largest square number factor, we get √975 = (√25)(√39) = 5√39 ≈ 31.22499

 

 

974 and Level 4

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This puzzle has ten clues and, like always, it has only one solution. If you can figure out where to put the factors 1 to 10 in the first column as well as the top row, then you will have found that solution.

Print the puzzles or type the solution in this excel file: 10-factors-968-977

This is my 974th post.

974 is the sum of three consecutive square numbers:
17² + 18² + 19² = 974

  • 974 is a composite number.
  • Prime factorization: 974 = 2 × 487
  • 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 974 has exactly 4 factors.
  • Factors of 974: 1, 2, 487, 974
  • Factor pairs: 974 = 1 × 974 or 2 × 487
  • 974 has no square factors that allow its square root to be simplified. √974 ≈ 31.20897

 

973 and Level 3

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If this were a Find the Factors 1 – 12 puzzle, then clues 72 and 24 would have three different possible common factors, but that isn’t the kind of puzzle it is. It is just a Find the Factors 1 – 10 puzzle, and there is just one possible common factor that will put only factors from 1 – 10 in the first column and the top row. Do you know that common factor? If you do, you will be well on your way to solving the puzzle!

Print the puzzles or type the solution in this excel file: 10-factors-968-977

Here are a few facts about the number 973:

It is palindrome 191 in BASE 27 because 1(27²) + 9(27) + 1(1) = 973

  • 973 is a composite number.
  • Prime factorization: 973 = 7 × 139
  • 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 973 has exactly 4 factors.
  • Factors of 973: 1, 7, 139, 973
  • Factor pairs: 973 = 1 × 973 or 7 × 139
  • 973 has no square factors that allow its square root to be simplified. √973 ≈ 31.1929

972 Happy Birthday, Andy!

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Today is my brother Andy’s birthday. I know Andy can solve these puzzles so I’ve made him a puzzle cake with factors from 1 to 16. Adding extra factor possibilities complicates the puzzle and makes it a little more difficult to read as a multiplication table, but it is still solvable. Since these puzzles have only one solution and are solved by logic and not by guessing and checking, I added a clue right in the center of the cake to ensure a unique solution. Happy birthday, Andy!

Print the puzzles or type the solution in this excel file: 10-factors-968-977

Now I’ll share a little about the number 972 which is the 13th Achilles number.  All of the exponents in its prime factorization are greater than 1, yet the greatest common factor of those exponents is still 1. The previous Achilles number, 968, and 972 are the closest two Achilles numbers so far.

I think 972 has some interesting representations when written in some other bases:

It’s 33030 in BASE 4 because 3(4⁴) + 2(4³) + 0(4²) + 3(4) + 0(1) = 3(256 + 64 + 4) = 3(324) = 972
363 in BASE 17 because 3(17²) + 6(17) + 3(1) = 972
300 in BASE 18 because 3(18²) = 3(324) = 972
RR in BASE 35 (R is 27 base 10) because 27(35) + 27(1) = 27(36) = 972
R0 in BASE 36 because 27(36) + 0(1) = 27(36) = 972

  • 972 is a composite number.
  • Prime factorization: 972 = 2 × 2 × 3 × 3 × 3 × 3 × 3, which can be written 972 = 2²× 3⁵
  • The exponents in the prime factorization are 2 and 5. Adding one to each and multiplying we get (2 + 1)(5 + 1) = 3 × 6 = 18. Therefore 972 has exactly 18 factors.
  • Factors of 972: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 243, 324, 486, 972
  • Factor pairs: 972 = 1 × 972, 2 × 486, 3 × 324, 4 × 243, 6 × 162, 9 × 108, 12 × 81, 18 × 54 or 27 × 36
  • Taking the factor pair with the largest square number factor, we get √972 = (√324)(√3) = 18√3 ≈ 31.1769

Here are a few of the MANY possible factor trees for 972:

970 and Level 1

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This level 1 puzzle will help you focus on one set of division facts. You can find all the factors that belong in the first column and the top row if you know those division facts. After you find all the factors from 1 to 10, you can fill in the entire multiplication table.

Print the puzzles or type the solution in this excel file: 10-factors-968-977

970 is the sum of two squares two different ways:
23² + 21² = 970
31² + 3²= 970

That means 970 is the hypotenuse of more than one Pythagorean triple:
88-966-970 calculated from 23² – 21², 2(23)(21), 23² + 21²
186-952-970 calculated from 2(31)(3), 31² – 3², 31² + 3²
582-776-970 which is (3-4-5) times 194
650-720-970 which is 10 times (65-72-97)

Here’s a fun fact: 970 is 202 in BASE 22 because 2(22²) + 2(1) = 2(484 + 1) = 2(485) = 970

  • 970 is a composite number.
  • Prime factorization: 970 = 2 × 5 × 97
  • 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 970 has exactly 8 factors.
  • Factors of 970: 1, 2, 5, 10, 97, 194, 485, 970
  • Factor pairs: 970 = 1 × 970, 2 × 485, 5 × 194, or 10 × 97
  • 970 has no square factors that allow its square root to be simplified. √970 ≈ 31.14482

 

969 is the 17th Tetrahedral Number and the 17th Nonagonal Number

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A tetrahedron is a pyramid whose base and sides are all triangles.

The nth tetrahedral number is the sum of the first n triangular numbers. So if you made a pyramid of the first n triangular numbers, you would get the nth triangular pyramidal number, also known as the nth tetrahedral number.

969 is the 17th tetrahedral number.

That image might look a little like a Christmas tree lot where you could select a tree in several different sizes. If we had tiny cubes instead of squares, we could stack them on top of each other to make a tetrahedron. That is the visual reason why 969 is a tetrahedron.

Look at the graphic below of a portion of Pascal’s triangle. You can easily see the first 19 counting numbers. The first 18 triangular numbers are highlighted in red, and the first 17 tetrahedral numbers are highlighted in green. The 16th tetrahedral number, 816, plus the 17th triangular number, 153, equals 969.

Because of its spot on Pascal’s triangle, I know that (17·18·19)/(1·2·3) = 969. That is the algebraic reason 969 is a tetrahedral number.

969 is also the 17th nonagonal number because 17(7·17 – 5)/2 = 969. I am not going to try to illustrate a 9-sided figure, but I’m sure it would be a cool image if I could.

All of this means that 969 is the 17th tetrahedral number AND the 17th nonagonal number. 1 is the smallest number to be both a tetrahedral number and a nonagonal number. 969 is the next smallest number to be both. Amazingly, it is the 17th of both, too!

969 obviously is a palindrome in base 10.

In base 20, it is 289. I find that quite curious because 17² = 289, and 17 is a factor of 969. Why would we write this number as 289 in base 20? Because 2(20²) + 8(20) + 9(1) = 969

Because 17 is one of its factors, 969 is the hypotenuse of a Pythagorean triple:
456-855-969 which is 57 times (8-15-17)

  • 969 is a composite number.
  • Prime factorization: 969 = 3 × 17 × 19
  • 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 969 has exactly 8 factors.
  • Factors of 969: 1, 3, 17, 19, 51, 57, 323, 969
  • Factor pairs: 969 = 1 × 969, 3 × 323, 17 × 57, or 19 × 51
  • 969 has no square factors that allow its square root to be simplified. √969 ≈ 31.12876