944 and Level 3

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The division facts needed to solve today’s puzzle are not complicated. You can fill in all the cells of this puzzle if you know the multiplication facts from 1 × 1 to 12 × 12.

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

Now here are some facts about the number 944:

944 is divisible by 2 because it is even.
944 is divisible by 4 because the last number is divisible by 4 and the digit before it is even.
944 can be evenly divided by 8 because 44 is divisible by 4, but not by 8, and the digit before 44 is odd.

944 is a funny-looking palindrome, 1I1, in BASE 23 (I is 18 in base 10) because 1(23²) + 18(23¹) + 1(23⁰) = 944

  • 944 is a composite number.
  • Prime factorization: 944 = 2 × 2 × 2 × 2 × 59, which can be written 944 = 2⁴ × 59
  • 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 944 has exactly 10 factors.
  • Factors of 944: 1, 2, 4, 8, 16, 59, 118, 236, 472, 944
  • Factor pairs: 944 = 1 × 944, 2 × 472, 4 × 236, 8 × 118, or 16 × 59
  • Taking the factor pair with the largest square number factor, we get √944 = (√16)(√59) = 4√59 ≈ 30.72458

943 and Level 2

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Level 2 puzzles aren’t very tricky, but maybe this one is a little bit. Can you write the factors from 1 to 12 in both the first column and the top row so that this puzzle functions as a multiplication table?

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

Now I’ll write something about the number 943:

943 is the hypotenuse of a Pythagorean triple:
207-920-943 which is 23 times (9-40-41)

It is also a leg in two primitive Pythagorean triples:
576-943-1105, calculated from 2(32)(9), 32² – 9², 32² + 9²
943-444624-444625, calculated from 472² – 471², 2(472)(471), 472² + 471²

  • 943 is a composite number.
  • Prime factorization: 943 = 23 × 41
  • 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 943 has exactly 4 factors.
  • Factors of 943: 1, 23, 41, 943
  • Factor pairs: 943 = 1 × 943 or 23 × 41
  • 943 has no square factors that allow its square root to be simplified. √943 ≈ 30.708305

942 and Level 1

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This puzzle is probably as tough as a level 1 puzzle can get, but don’t let that prevent you from giving it a try! Can you figure out where the factors from 1 to 12 go in both the first column and the top row?

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

Now let me tell you a little about the number 942:

It is the sum of four consecutive prime numbers:
229 + 233 + 239 + 241 = 942

It is the hypotenuse of a Pythagorean triple:
510-792-942 which is 6 times (85-132-157)

942 is a palindrome in two other bases and a repdigit in another:
787 in BASE 11, because 7(11²) + 8(11¹) + 7(11⁰) = 942
272 in BASE 20, because 2(20²) + 7(20¹) + 2(20⁰) = 942
666 in BASE 12, because 6(12²) + 6(12¹) + 6(12⁰) = 6(144+12+1) = 6(157) = 942

942³ is 835,896,888. OEIS.org tells us that 942³ is the smallest perfect cube that contains five 8‘s.

  • 942 is a composite number.
  • Prime factorization: 942 = 2 × 3 × 157
  • 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 942 has exactly 8 factors.
  • Factors of 942: 1, 2, 3, 6, 157, 314, 471, 942
  • Factor pairs: 942 = 1 × 942, 2 × 471, 3 × 314, or 6 × 157
  • 942 has no square factors that allow its square root to be simplified. √942 ≈ 30.6920185

941 and Level 6

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Today daylight savings time ends in the United States. That gives you an extra hour to sleep or do whatever you want. Perhaps you could spend part of your extra hour solving today’s puzzle:

Print the puzzles or type the solution on this excel file: 10-factors-932-941

941 is a prime number that is also the sum of consecutive prime numbers two different ways:
179 + 181 + 191 + 193 + 197 = 941; that’s five consecutive primes
311 + 313 + 317 = 941; that’s three consecutive primes

29² + 10² = 941 That means 941 is the hypotenuse of a Pythagorean triple:
580-741-941 which can be calculated from 2(29)(10), 29² – 10², 29² + 10²

941 is also palindrome 575 in BASE 13 because 5(13²) + 7(13¹) + 5(13⁰) = 941

  • 941 is a prime number.
  • Prime factorization: 941 is prime.
  • The exponent of prime number 941 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 941 has exactly 2 factors.
  • Factors of 941: 1, 941
  • Factor pairs: 941 = 1 × 941
  • 941 has no square factors that allow its square root to be simplified. √941 ≈ 30.6757233

How do we know that 941 is a prime number? If 941 were not a prime number, then it would be divisible by at least one prime number less than or equal to √941 ≈ 30.7. Since 941 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 941 is a prime number.

Here’s another way we know that 941 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 29² + 10² = 941 with 29 and 10 having no common prime factors, 941 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √941 ≈ 30.7. Since 941 is not divisible by 5, 13, 17, or 29, we know that 941 is a prime number.

 

940 Mystery Level

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Today’s puzzle reminds me of a gumball machine. I would invite you to stick to solving this puzzle until you find success. I assure you that the factors from 1 to 10 can be placed in the first column and the top row solely by using logic.

Print the puzzles or type the solution on this excel file: 10-factors-932-941

Now let me tell you something about the number 940.

940 is the hypotenuse of a Pythagorean triple:
564-752-940 which is (3-4-5) times 188

  • 940 is a composite number.
  • Prime factorization: 940 = 2 × 2 × 5 × 47, which can be written 940 = 2² × 5 × 47
  • 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 940 has exactly 12 factors.
  • Factors of 940: 1, 2, 4, 5, 10, 20, 47, 94, 188, 235, 470, 940
  • Factor pairs: 940 = 1 × 940, 2 × 470, 4 × 235, 5 × 188, 10 × 94, or 20 × 47,
  • Taking the factor pair with the largest square number factor, we get √940 = (√4)(√235) = 2√235 ≈ 30.659419

Is There Anything Else Special about the Palindrome 939?

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Yes, 939 is a palindrome in base 10, but also all of its factors (1, 3, 313, and 939) are palindromes. It is also palindrome
32223 in BASE 4 because 3(4⁴) + 2(4³) + 2(4²) + 2(4¹) + 3(4⁰) = 939

Okay, that’s nice. Is there anything else special about 939?

The first ten decimal places of the cube root of 939 contain ALL ten digits 0 to 9. That’s unusual, and a reason why 939 is a special number. I made this gif to highlight its uniqueness.
Cube Root 939

make science GIFs like this at MakeaGif

***********

Thank you OEIS.org for informing us of that amazing fact about 939’s cube root.

939 is also the hypotenuse of a Pythagorean triple:
75-936-939 which is 3 times (25-312-313)

  • 939 is a composite number.
  • Prime factorization: 939 = 3 × 313
  • 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 939 has exactly 4 factors.
  • Factors of 939: 1, 3, 313, 939
  • Factor pairs: 939 = 1 × 939 or 3 × 313
  • 939 has no square factors that allow its square root to be simplified. √939 ≈ 30.64310689

 

938 and Level 5

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Can you figure out where the factors 1 – 10 go in the first column and top row so that this level 5 puzzle will function as a multiplication table?

Print the puzzles or type the solution on this excel file: 10-factors-932-941

Now I’ll share a few facts about the number 938.

938 is a palindrome in two consecutive bases:
It’s 343 in BASE 17 because 3(17²) + 4(17¹) + 3(17º) = 938
It’s 2G2 in BASE 18 (G is 16 base 10), because 2(18²) + 16(18¹) + 2(18º) = 938

  • 938 is a composite number.
  • Prime factorization: 938 = 2 × 7 × 67
  • 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 938 has exactly 8 factors.
  • Factors of 938: 1, 2, 7, 14, 67, 134, 469, 938
  • Factor pairs: 938 = 1 × 938, 2 × 469, 7 × 134, or 14 × 67
  • 938 has no square factors that allow its square root to be simplified. √938 ≈ 30.62678566

Why Is Prime Number 937 the 13th Star Number?

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Even though 937 is a prime number, 937 tiny rectangles can be arranged into this beautiful star. Why?

937 is the 13th star number because 6(13)(13 – 1) + 1 = 937.

It is also the 13th star number because it is 12 times the 12th triangular number plus one: Look at this pattern:

The first star number is 12 times the 0th triangular number plus 1. Thus, 12(0) + 1 = 1 (1 yellow rectangle in the center)
The second star number is 12 times the 1st triangular number plus 1. Thus, 12(1) + 1 = 13 (12 green + 1 yellow rectangle in the center)
The third star number is 12 times the 2nd triangular number plus 1. Thus, 12(3) + 1 = 37 (24 blue + 12 green + 1 yellow rectangle in the center)
and so on. . .until
The thirteen star number is 12 times the 12th triangular number plus 1. Thus, 12(78) + 1 = 937 (144 yellow + 132 orange + 120 red + 108 purple + 96 blue + 84 green + 72 yellow + 60 orange + 48 red + 36 purple + 24 blue + 12 green + 1 yellow rectangle in the center)

I made the star so that it consists of one tiny rectangle in the center surrounded by 6 triangles with 78 (the 12th triangular number) rectangles each with another 6 triangles of the same size to form the 6 points of the star.

I very much enjoyed making this star. If you look closely you will see thirteen concentric stars in it following the pattern yellow, green, blue, purple, red, and orange repeated. I added star outlines to make the three smallest stars easier to see.

I think the graphic says a lot about the number 937 all by itself. I hope you enjoy looking at it.

Here’s a little more about the number 937:

24² + 19² = 937, so 937 is the hypotenuse of a Pythagorean triple:
215-912-937 which can be calculated from 24² – 19², 2(24)(19), 24² + 19²

937 is also a palindrome in three other bases:
1021201 in BASE 3 because 1(3⁶) + 0(3⁵) + 2(3⁴) + 1(3³) + 2(3²) + 0(3¹) + 1(3⁰) = 937
1F1 in BASE 24 (F is 15 in base 10) because 1(24²) + 15(24¹) + 1(24⁰) = 937
1A1 in BASE 26 (A is 10 in base 10) because 1(26²) + 10(26¹) + 1(26⁰) = 937

  • 937 is a prime number.
  • Prime factorization: 937 is prime.
  • The exponent of prime number 937 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 937 has exactly 2 factors.
  • Factors of 937: 1, 937
  • Factor pairs: 937 = 1 × 937
  • 937 has no square factors that allow its square root to be simplified. √937 ≈ 30.6104557

How do we know that 937 is a prime number? If 937 were not a prime number, then it would be divisible by at least one prime number less than or equal to √937 ≈ 30.6. Since 937 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 937 is a prime number.

Here’s another way we know that 937 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 24² + 19² = 937 with 24 and 19 having no common prime factors, 937 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √937 ≈ 30.6. Since 937 is not divisible by 5, 13, 17, or 29, we know that 937 is a prime number.

936 and Level 4

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Will this puzzle be smooth sailing for you? You won’t know until you give it a try!

Print the puzzles or type the solution on this excel file: 10-factors-932-941

936 is the 12th pentagonal pyramidal number because 12²(12 + 1)/2 = 936.

936 is the sum of the twelve prime numbers from 53 to 103.

30² + 6² = 936, so 936 is the hypotenuse of a Pythagorean triple:
360-864-936 which is (5-12-13) times 72, and
can be calculated from 2(30)(6), 30² – 6², 30² + 6²

936 is palindrome 12221 in BASE 5 because 1(5⁴) + 2(5³) + 2(5²) + 2(5¹) + 1(5º) = 936
and palindrome QQ in BASE 35 (Q is 26 base 10), because 26(35) + 26(1) = 26(36) = 936
It is also Q0 in BASE 36 because 26(36) + 0(1) = 936

  • 936 is a composite number.
  • Prime factorization: 936 = 2 × 2 × 2 × 3 × 3 × 13, which can be written 936 = 2³ × 3² × 13
  • The exponents in the prime factorization are 3, 2 and 1. Adding one to each and multiplying we get (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24. Therefore 936 has exactly 24 factors.
  • Factors of 936: 1, 2, 3, 4, 6, 8, 9, 12, 13, 18, 24, 26, 36, 39, 52, 72, 78, 104, 117, 156, 234, 312, 468, 936
  • Factor pairs: 936 = 1 × 936, 2 × 468, 3 × 312, 4 × 234, 6 × 156, 8 × 117, 9 × 104, 12 × 78, 13 × 72, 18 × 52, 24 × 39¸ or 26 × 36
  • Taking the factor pair with the largest square number factor, we get √936 = (√36)(√26) = 6√26 ≈ 30.594117.

935 Is the Second Lucas-Carmichael Number

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935 = 5 × 11 × 17, and 935 + 1 is evenly divisible by 5 + 1, 11 + 1, and 17 + 1. That makes 935 only the SECOND Lucas-Carmichael number. Thanks to OEIS.org for that fun fact.

Today’s puzzle is a level 3, a good transition from the easier puzzles to the more difficult ones.

Print the puzzles or type the solution on this excel file: 10-factors-932-941

Here’s more about the number 935:

935 is the sum of the nineteen prime numbers from 13 to 89.

935 is the hypotenuse of four Pythagorean triples:
143-924-935, which is 11 times (13-84-85)
396-847-935, which is 11 times (36-77-85)
440-825-935, which is (8-15-17) times 55
561-748-935, which is (3-4-5) times 187

  • 935 is a composite number.
  • Prime factorization: 935 = 5 × 11 × 17
  • 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 935 has exactly 8 factors.
  • Factors of 935: 1, 5, 11, 17, 55, 85, 187, 935
  • Factor pairs: 935 = 1 × 935, 5 × 187, 11 × 85, or 17 × 55
  • 935 has no square factors that allow its square root to be simplified. √935 ≈ 30.5777697