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:

971 and Level 2

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Some of the clues in today’s puzzles are perfect squares. Some aren’t. Can you figure out which are which and put their factors in the right places in the first column and the top row?

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

971 is palindrome 2D2 in BASE 19 (D is 13 base 10)
because 2(19²) + 13(19) + 2(1) = 971

  • 971 is a prime number.
  • Prime factorization: 971 is prime.
  • The exponent of prime number 971 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 971 has exactly 2 factors.
  • Factors of 971: 1, 971
  • Factor pairs: 971 = 1 × 971
  • 971 has no square factors that allow its square root to be simplified. √971 ≈ 31.16087

How do we know that 971 is a prime number? If 971 were not a prime number, then it would be divisible by at least one prime number less than or equal to √971 ≈ 31.2. Since 971 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 971 is a prime number.

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

 

 

 

Reasons to Celebrate 968

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9 – 6 + 8 = 11, so 968 is divisible by 11. In fact, 11 is its largest prime factor so we can make a beautiful factor cake for 968 with two candles on top.

Guess what, 968 is also divisible by 11², so its factor cake can have even more candles!

Look at its factor cake. Notice that each of 968’s prime factors is repeated at least once. 2³ × 11² = 968

OEIS.org alerts us to the fact that 968 is the twelfth Achilles number. That means that each of its prime factors has an exponent greater than one yet the greatest common factor of those exponents is still one. (Perfect squares, cubes, etc. are not Achilles numbers.)

There are thirteen Achilles numbers less than 1000. Here is a chart of them and their prime factorizations. These numbers appear to be few and far between. The previous Achilles number was 104 less than 968, but the next one is only 4 numbers away!

Being only 4 numbers away is pretty amazing. Consecutive Achilles numbers actually exist. You can find the smallest pair of them in the Wikipedia article. Both numbers are greater than 5 billion. Again, being only 4 numbers away is pretty amazing.

The smallest Achilles number made with three different prime numbers raised to various powers is 2³·3²·5² = 1800. Notice that each exponent is greater than one yet the greatest common factor of those exponents is still one.

Here’s a little more about the number 968:

I like the way 968 looks in a few other bases:
It’s 2552 in BASE 7 because 2(7³) + 5(7²) + 5(7¹) + 2(7⁰) = 968,
800 in BASE 11 because 8(11²) = 8(121) = 968,
242 in BASE 21 because 2(21²) + 4(21¹) + 2(21⁰) = 968,
200 in BASE 22 because 2(22²) = 2(484) = 968.

  • 968 is a composite number.
  • Prime factorization: 968 = 2 × 2 × 2 × 11 × 11, which can be written 968 = 2³ × 11²
  • The exponents in the prime factorization are 3 and 2. Adding one to each and multiplying we get (3 + 1)(2 + 1) = 4 × 3 = 12. Therefore 968 has exactly 12 factors.
  • Factors of 968: 1, 2, 4, 8, 11, 22, 44, 88, 121, 242, 484, 968
  • Factor pairs: 968 = 1 × 968, 2 × 484, 4 × 242, 8 × 121, 11 × 88, or 22 × 44
  • Taking the factor pair with the largest square number factor, we get √968 = (√484)(√2) = 22√2 ≈ 31.11269837

 

967 Black Friday Shopping Advantage

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If your shopping cart were a go-kart, you would have an advantage getting all the shopping bargains Black Friday offers. Not only would you be able to move much faster than the average shopping cart, but you would also be able to do wheelies to get through the crowds, around corners, or tight spaces. After you complete the shopping spree of your dreams, you can lie down exhausted, but ecstatic and work on a puzzle, like this one.

Print the puzzles or type the solution in this excel file: 12 factors 959-967

I realize I’m really pushing it to make this puzzle have a Thanksgiving week theme. I love that Black Friday has turned into Black November because it means bargains without all the crowds.

You can also imagine the puzzle is a toy on a child’ wishlist. Whatever you think, I hope you enjoy solving the puzzle.

Here’s a little about prime number 967:

It is 595 in BASE 13 because 5(13²) + 9(13¹) + 5(13⁰) = 967
It is also 1J1 in BASE 23 (J is 19 in base 10) because 1(23²) + 19(23¹) + 1(23⁰) = 967

  • 967 is a prime number.
  • Prime factorization: 967 is prime.
  • The exponent of prime number 967 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 967 has exactly 2 factors.
  • Factors of 967: 1, 967
  • Factor pairs: 967 = 1 × 967
  • 967 has no square factors that allow its square root to be simplified. √967 ≈ 31.09662

How do we know that 967 is a prime number? If 967 were not a prime number, then it would be divisible by at least one prime number less than or equal to √967 ≈ 31.1. Since 967 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, or 31, we know that 967 is a prime number.

966 Groan! Gotta Loosen My Belt

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If you’ve overeaten this Thanksgiving day, you may be in too much pain to start working off all those extra calories. You may just want to loosen your belt and lie down somewhere while you groan about eating so much. Exercising your brain may help you alleviate some of that regret. This puzzle with its Pilgrim belt buckle could be just what you need. It’s a level 6 so it won’t be easy, but you will feel very accomplished if you can solve it.

Print the puzzles or type the solution in this excel file: 12 factors 959-967

Here’s a little information about the number 966:

It is the sum of eight consecutive prime numbers:
103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 = 966

It is also the sum of two consecutive prime numbers:
467 + 479 = 966

966 is palindrome 686 in BASE 12 because 6(144) + 8(12) + 6(1) = 966

  • 966 is a composite number.
  • Prime factorization: 966 = 2 × 3 × 7 × 23
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 966 has exactly 16 factors.
  • Factors of 966: 1, 2, 3, 6, 7, 14, 21, 23, 42, 46, 69, 138, 161, 322, 483, 966
  • Factor pairs: 966 = 1 × 966, 2 × 483, 3 × 322, 6 × 161, 7 × 138, 14 × 69, 21 × 46, or 23 × 42
  • 966 has no square factors that allow its square root to be simplified. √966 ≈ 31.08054

965 Run, Turkey, Run!

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Happy Thanksgiving, everyone!

I didn’t mean to make any Thanksgiving puzzles but after I created the puzzles this week, I could see some Thanksgiving-related pictures in the designs I had already made. This one is my favorite.

Run, Turkey, Run! For millions of turkeys today, it’s already too late.

Print the puzzles or type the solution in this excel file: 12 factors 959-967

Here is a little about the number 965:

965 is the sum of two squares two different ways:
26² + 17² = 965
31² + 2² = 965

So it is also the hypotenuse of FOUR Pythagorean triples, two of them primitives:
124-957-965, calculated from 2(31)(2), 31² – 2², 31² + 2²
387-884-965, calculated from 26² – 17², 2(26)(17), 26² + 17²
475-840-965 which is 5 times (95-168-193)
579-772-965 which is (3-4-5) times 193

  • 965 is a composite number.
  • Prime factorization: 965 = 5 × 193
  • 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 965 has exactly 4 factors.
  • Factors of 965: 1, 5, 193, 965
  • Factor pairs: 965 = 1 × 965 or 5 × 193
  • 965 has no square factors that allow its square root to be simplified. √965 ≈ 31.064449

964 Bow and Arrow

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You can google lots of images of bows and arrows related to Thanksgiving. Perhaps that is because Indians from the Wampanoag Nation joined the Pilgrims in what we call the first Thanksgiving. I’m sure you know the story very well told from the white man’s point of view. It will be well worth your time to read The REAL Story of Thanksgiving as well. Everyone’s History Matters is also an excellent read.

Print the puzzles or type the solution in this excel file: 12 factors 959-967

 

Now I’ll tell you a little about the number 964:

964 is the sum of the fourteen prime numbers from 41 to 101.

It is also the sum of four consecutive prime numbers:
233 + 239 + 241 + 251 = 964

30² + 8²  = 964 making it the hypotenuse of a Pythagorean triple:
480-836-964 which is 4 times (120-209-241)

964 is a palindrome in 2 bases and a repdigit in another.
1022201 in BASE 3
7A7 in BASE 11 (A is 10 in base 10)
444 in BASE 15

  • 964 is a composite number.
  • Prime factorization: 964 = 2 × 2 × 241, which can be written 964 = 2² × 241
  • 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 964 has exactly 6 factors.
  • Factors of 964: 1, 2, 4, 241, 482, 964
  • Factor pairs: 964 = 1 × 964, 2 × 482, or 4 × 241
  • Taking the factor pair with the largest square number factor, we get √964 = (√4)(√241) = 2√241 ≈ 31.048349

963 Arrow

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The numbers 1 to 12 fit someplace in the first column as well as in the top row.  Can you figure out where those places are so that this puzzle can become a multiplication table?

Print the puzzles or type the solution in this excel file: 12 factors 959-967

Now I’d like to mention a few things about the number 963:

It is the sum of the 24 prime numbers from 2 to 89. Pretty cool!

I like the way 963 looks in base 10 as well as in a few other bases:
33003 in BASE 4 because 3(4⁴) + 3(4³) + 3(1) = 3 × 321 = 963
3C3 in BASE 16 (C is 12 in base 10) because 3(16²) + 12(16) + 3(1) = 963
1B1 in BASE 26 (B is 11 in base 10) because 1(26²) + 11(26) + 1(1) = 963
123 in BASE 30 because 1(30²) + 2(30¹) + 3(30⁰) = 963

  • 963 is a composite number.
  • Prime factorization: 963 = 3 × 3 × 107, which can be written 963 = 3² × 107
  • 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 963 has exactly 6 factors.
  • Factors of 963: 1, 3, 9, 107, 321, 963
  • Factor pairs: 963 = 1 × 963, 3 × 321, or 9 × 107
  • Taking the factor pair with the largest square number factor, we get √963 = (√9)(√107) = 3√107 ≈ 31.03224