A Multiplication Based Logic Puzzle

Posts tagged ‘Factors’

964 Bow and Arrow

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.

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

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963 Arrow

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

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

961 is a Perfect Square in More Ways Than One

961 is a perfect square. Its square root is 31. Look what happens when their digits are reversed. We still get a perfect square!

The digits 1, 6, and 9 form six different 3-digit numbers when each of those digits is used only one time. Here are some square facts about each of those numbers:

169 = 13²
196 = 14²
619 = 310² – 309²
691 = 346² – 345²
916 = 30² + 4²
961 = 31²

Of course, 169 also equals 5² + 12²

So 961 is a perfect square forward, backward, and upside-down!

My friend, Muthu Yuvaraj, shared a couple of similar square facts with me:

and

Fascinating!

Here’s more information about perfect square 961:

It is the sum of three consecutive square numbers:
313 + 317 + 331 = 961

It is the sum of five consecutive square numbers:
181 + 191 + 193 + 197 + 199 = 961

And it is the sum of the twenty-three prime numbers from 3 to 89.

It is palindrome 12321 in BASE 5 because 1(5⁴) + 2(5³) + 3(5²) + 2(5¹) + 1(5⁰) = 961

961 is a perfect square in base 10, and it looks like a perfect square in some other bases, too:
100 in BASE 31 because 1(31²) = 961
121 in BASE 30 because 1(30²) + 2(30¹) + 1(30⁰) = 961
144 in BASE 29 because 1(29²) + 4(29¹) + 4(29⁰) = 961
169 in BASE 28 because 1(28²) + 6(28¹) + 9(28⁰) = 961
441 in BASE 15 because 4(15²) + 4(15¹) + 1(15⁰) = 961
Remarkably, we saw 961, 169, 144, and 441 already in graphics for this post! So here’s one more graphic:

  • 961 is a composite number.
  • Prime factorization: 961 = 31²
  • The exponent in the prime factorization is 2. Adding one we get (2 + 1) = 3. Therefore 961 has exactly 3 factors.
  • Factors of 961: 1, 31, 961
  • Factor pairs: 961 = 1 × 961 or 31 × 31
  • 961 is a perfect square. √961 = 31

960 Factor Trees

960 is the smallest number to have exactly 28 factors. 960 is 2⁶·3·5, so any factor tree made for it will have 6 + 1 + 1 = 8 prime factors. Since 8 is a power of 2, this number, 960, has some beautiful and well-balanced factor trees as well as some that aren’t as good-looking. Here are five of the MANY possible factor trees for 960:

 

960 can be written as the difference of 2 squares TEN different ways:

  1. 241² – 239² = (241 + 239)(241 – 239) = 480 × 2 = 960
  2. 122² – 118² = (122 + 118)(122 – 118) = 240 × 4 = 960
  3. 83² – 77² = (83 + 77)(83 – 77) = 160 × 6 = 960
  4. 64² – 56² = (64 + 56)(64 – 56) = 120 × 8 = 960
  5. 53² – 43² = (53 + 43)(53 – 43) = 96 × 10 = 960
  6. 46² – 34² = (46 + 34)(46 – 34) = 80 × 12 = 960
  7. 38² – 22² = (38 + 22)(38 – 22) = 60 × 16 = 960
  8. 34² – 14² = (34 + 14)(34 – 14) = 48 × 20 = 960
  9. 32² – 8² = (32 + 8)(32 – 8) = 40 × 24 = 960
  10. 31² – 1² = (31 + 1)(31 – 1) = 32 × 30 = 960

960 is the sum of the sixteen prime numbers from 29 to 97.

It is also the sum of six consecutive prime numbers:
149 + 151 + 157 + 163 + 167 + 173 = 960

960 is the hypotenuse of a Pythagorean triple:
576-768-960 which is (3-4-5) times 192

I like how 960 looks in these other bases:
33000 in BASE 4 because 3(4⁴) + 3(4³) = 3(256 + 64) = 3 × 320 = 960
440 in BASE 15 because 4(15²) + 4(15) = 4(225 + 15) = 4 × 240 = 960
UU in BASE 31 (U is 30 base 10), because 30(31) + 30(1) = 30(31 + 1) = 30 × 32 = 960
U0 in BASE 32 because 30(32) + 0 = 960

Stetson.edu informs us that 9 + 6 + 09³ + 6³ + 0³ = 960

  • 960 is a composite number and a perfect square.
  • Prime factorization: 960 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 5, which can be written 960 = 2⁶ × 3 × 5
  • The exponents in the prime factorization are 6, 1 and 1. Adding one to each and multiplying we get (6 + 1)(1 + 1)(1 + 1) = 7 × 2 × 2 = 28. Therefore 960 has exactly 28 factors.
  • Factors of 960: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, 80, 96, 120, 160, 192, 240, 320, 480, 960
  • Factor pairs: 960 = 1 × 960, 2 × 480, 3 × 320, 4 × 240, 5 × 192, 6 × 160, 8 × 120, 10 × 96, 12 × 80, 15 × 64, 16 × 60, 20 × 48, 24 × 40, or 30 × 32
  • Taking the factor pair with the largest square number factor, we get √960 = (√64)(√15) = 8√15 ≈ 30.9838668

959 and Level 1

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

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

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