Puzzle

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

962 Buckle Up, Pilgrim

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Perhaps the most distinctive element of the pilgrims’ wardrobe is the buckle. We see it around the men’s waists, on their shoes, and on their hats. At least we do we if are looking at pilgrim costumes. It probably didn’t play as prominent a role on their actual clothes. Nevertheless, it is featured here on today’s puzzle. So buckle up and see where this puzzle takes you.

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

Here are a few facts about the number 962:

It is the sum of two squares two different ways:
31² +  1² = 962
29² +  11² =  962

It is also the hypotenuse of four Pythagorean triples:
62-960-962, calculated from 2(31)(1), 31² –  1², 31² +  1²
312-910-962, which is 2 times (156-455-481)
370-888-962, which is 2 times (185-444-481)
638-720-962, calculated from 2(29)(11), 29² –  11², 29² +  11²

It looks interesting in a few other bases:
4242 BASE 6
282 BASE 20
101 BASE 31
QQ BASE 36 (Q is 26 in base 10)

  • 962 is a composite number.
  • Prime factorization: 962 = 2 × 13 × 37
  • 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 962 has exactly 8 factors.
  • Factors of 962: 1, 2, 13, 26, 37, 74, 481, 962
  • Factor pairs: 962 = 1 × 962, 2 × 481, 13 × 74, or 26 × 37
  • 962 has no square factors that allow its square root to be simplified. √962 ≈ 31.01612

959 and Level 1

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Whether you are young, old, or in between, if you can do some simple division, then you can solve this level 1 puzzle. There is a column of clues and a row of clues. Both of them have the same common factor. Write that common factor in the first column to the left of the row of clues and again in the top row above the column of clues. Then simply divide. You will be done in no time at all.

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

Here are some facts about the number 957:

959 is the hypotenuse of a Pythagorean triple:
616-735-959 which is 7 times (88-105-137)

959 is a palindrome in base 10.

And it is a cool-looking 1110111111 in BASE 2
because (2¹⁰ – 1) – 2⁶ = 959.
In base 2 we would write (if we use commas)
1,111,111,111 – 1,000,000 = 1,110,111,111

  • 959 is a composite number.
  • Prime factorization: 959 = 7 × 137
  • 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 959 has exactly 4 factors.
  • Factors of 959: 1, 7, 137, 959
  • Factor pairs: 959 = 1 × 959 or 7 × 137
  • 959 has no square factors that allow its square root to be simplified. √959 ≈ 30.967725

 

958 and Level 6

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This puzzle is a multiplication table. You don’t have to be fast to solve it, but you do have to think. There is only one solution. The ten clues given are sufficient to find the places to put the factors 1 to 10 in the first column and the top row. Good luck!

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

958 is the sum of the 22 prime numbers from 5 to 89.

958 is also 141 in BASE 29 because 1(29²) + 4(29¹) + 1(29⁰) = 958

  • 958 is a composite number.
  • Prime factorization: 958 = 2 × 479
  • 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 958 has exactly 4 factors.
  • Factors of 958: 1, 2, 479, 958
  • Factor pairs: 958 = 1 × 958 or 2 × 479
  • 958 has no square factors that allow its square root to be simplified. √958 ≈ 30.951575

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