954 and Level 3

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The ten clues in this puzzle are all that is needed to solve the puzzle and then make it a complete multiplication table. Seriously, ten clues! Yes, the factors and the products will not be in their usual places, but the completed puzzle will still be a legitimate multiplication table. Can YOU make it work?

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

954 is the sum of ten consecutive prime numbers:
73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 = 954

27² +  15² = 954 so 954 is the hypotenuse of a Pythagorean triple:
504-810-954 which is 18 times (28-45-53)

954 is a palindrome in two consecutive bases:
676 in BASE 12 because 6(144) + 7(12) + 6(1) = 954
585 in BASE 13 because 5(169) + 8(13) + 5(1) = 954

  • 954 is a composite number.
  • Prime factorization: 954 = 2 × 3 × 3 × 53, which can be written 954 = 2 × 3² × 53
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 954 has exactly 12 factors.
  • Factors of 954: 1, 2, 3, 6, 9, 18, 53, 106, 159, 318, 477, 954
  • Factor pairs: 954 = 1 × 954, 2 × 477, 3 × 318, 6 × 159, 9 × 106, or 18 × 53,
  • Taking the factor pair with the largest square number factor, we get √954 = (√9)(√106) = 3√106 ≈ 30.88689

953 and Level 2

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This level 2 puzzle is only a tiny bit more difficult than a level 1 puzzle is. Start by finding the common factors of 4, 12, 40, and 28. The common factors are 1, 2, and 4, but 4 is the only one that works for the puzzle because we aren’t allowed to put factors like 14 or 28 in the top row. We are only allowed to write factors from 1 to 10 in the first column or the top row. Give this puzzle a try. I’m confident you can solve it!

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

Prime number 953 is the sum of the 21 prime numbers from 7 to 89.

28² + 13² = 953, so 953 is the hypotenuse of a Pythagorean triple:
615-728-953 calculated from 28² – 13², 2(28)(13), 28² + 13²

953 is a palindrome in base 11 and base 28:
797 in BASE 11 because 7(11²) + 9(11¹) + 7(11⁰) = 953
161 in BASE 28 because 1(28²) + 6(28¹) + 1(28⁰) = 953

953 × 19 × 3 = 54321, making 953 its biggest prime factor. Thank you OEIS.org
for that fun fact.

  • 952 is a prime number.
  • Prime factorization: 952 is prime.
  • The exponent of prime number 952 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 952 has exactly 2 factors.
  • Factors of 952: 1, 952
  • Factor pairs: 952 = 1 × 952
  • 952 has no square factors that allow its square root to be simplified. √952 ≈ 30.854497

How do we know that 952 is a prime number? If 952 were not a prime number, then it would be divisible by at least one prime number less than or equal to √952 ≈ 30.9. Since 952 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 952 is a prime number.

Here’s another way we know that 952 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 28² + 13² = 952 with 28 and 13 having no common prime factors, 952 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √952 ≈ 30.9. Since 952 is not divisible by 5, 13, 17, or 29, we know that 952 is a prime number.

 

 

 

952 and Level 1

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If you’ve never solved a Find the Factors puzzle before, this one will be perfect for you to do. It only has nine clues, but that is enough to find all the factors and fill in the entire multiplication table. You will feel quite smart when you’re done, too.

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

This is my 952nd post so I will mention a few facts about that number.

OEIS.org informs us that 93 + 53 + 23 + 9 × 5 × 2 = 952.

952 is the hypotenuse of a Pythagorean triple:
448-840-952 which is (8-15-17) times 56

952 looks interesting in some other bases:
4224 in BASE 6 because 4(6³) + 2(6²) + 2(6¹) + 4(6⁰) = 952
2C2 in BASE 19 (C is 12 Base 10) because 2(19²) + 12(19¹) +2(19⁰) = 952
SS BASE 33 (S is 28) because 28(33¹) + 28(33⁰) = 28(33 + 1) = 28(34) = 952
S0 BASE 34 because 28(34¹) + 0(34⁰) = 28(34) = 952

  • 952 is a composite number.
  • Prime factorization: 952 = 2 × 2 × 2 × 7 × 17, which can be written 952 = 2³ × 7 × 17
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 952 has exactly 16 factors.
  • Factors of 952: 1, 2, 4, 7, 8, 14, 17, 28, 34, 56, 68, 119, 136, 238, 476, 952
  • Factor pairs: 952 = 1 × 952, 2 × 476, 4 × 238, 7 × 136, 8 × 119, 14 × 68, 17 × 56, or 28 × 34
  • Taking the factor pair with the largest square number factor, we get √952 = (√4)(√238) = 2√238 ≈ 29.854497

951 is the 20th Centered Pentagonal Number

/ ivasallay / 3 Comments

Since 951 is the 20th centered pentagonal number, I decided to make the following graphic with 20 concentric pentagons. I’ve outlined the pentagons in the center to make them clearer. The graphic also shows that 951 is one more than five times the 19th triangular number.

951 is also the hypotenuse of a Pythagorean triple:
225-924-951 which is 3 times (75-308-317)

As numbers get bigger, palindromes in base 2 get rarer, but 951 is one of them:
1110110111 in BASE 2 because 1(2⁹) + 1(2⁸) + 1(2⁷) + 0(2⁶) + 1(2⁵) + 1(2⁴) + 0(2³) + 1(2²) + 1(2¹) +1(2⁰) = 951
It is also 1D1 in BASE 25 (D is 14 base 10) because 1(25²) + 13(25¹) + 1(25⁰) = 951

  • 951 is a composite number.
  • Prime factorization: 951 = 3 × 317
  • 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 951 has exactly 4 factors.
  • Factors of 951: 1, 3, 317, 951
  • Factor pairs: 951 = 1 × 951 or 3 × 317
  • 951 has no square factors that allow its square root to be simplified. √951 ≈ 30.83828789

950 and Level 6

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Today’s puzzle is a level 6, so it isn’t for beginners. It is trickier than the easier levels, but don’t let that stop you from giving it a try. If you use guess and check, you will likely become frustrated, but you can solve this puzzle IF you use logic before you write each factor.

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

950 is the hypotenuse of two Pythagorean triples:
266-912-950 which is (7-24-25) times 38
570-760-950 which is (3-4-5) times 190

950 is 2525 in BASE 7 because 2(7³) + 5(7²) + 2(7¹) + 5(7⁰) = 950

  • 950 is a composite number.
  • Prime factorization: 950 = 2 × 5 × 5 × 19, which can be written 950 = 2 × 5² × 19
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 950 has exactly 12 factors.
  • Factors of 950: 1, 2, 5, 10, 19, 25, 38, 50, 95, 190, 475, 950
  • Factor pairs: 950 = 1 × 950, 2 × 475, 5 × 190, 10 × 95, 19 × 50, or 25 × 38,
  • Taking the factor pair with the largest square number factor, we get √950 = (√25)(√38) = 5√38 ≈ 30.82207

 

949 Mystery Level

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The difficulty level of a mystery level puzzle is a mystery. You won’t know how hard or how easy it is until you give it a try. You can solve it by using logic and your knowledge of the multiplication table. Can you figure it out?

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

Now here is a little about the number 949:

948 and 949 are a Ruth-Aaron pair.

949 is not only a palindrome in base 10, but it is also
434 in BASE 15 because 4(15²) + 3(15¹) + 4(15⁰) = 949

25² + 18² = 949 and 30² + 7² = 949 so 949 is the hypotenuse of four Pythagorean triples:

301-900-949 calculated from 25² – 18², 2(25)(18), 25² + 18²
365-876-949 which is (5-12-13) times 73
420-851-949 calculated from 2(30)(7), 30² – 7², 30² + 7²
624-715-949 which is 13 times (48-55-73)

  • 949 is a composite number.
  • Prime factorization: 949 = 13 × 73
  • 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 949 has exactly 4 factors.
  • Factors of 949: 1, 13, 73, 949
  • Factor pairs: 949 = 1 × 949 or 13 × 73
  • 949 has no square factors that allow its square root to be simplified. √949 ≈ 30.80584

948 and Level 5

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All you need is a little bit of logic and the multiplication facts in a standard 12 x 12 multiplication table to solve this puzzle. This puzzle is not just for kids. It can be challenging even for adults. Go ahead, give it a try. You’ll soon see that it’s tougher than it looks, but stick with it, you can conquer it!

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

Here’s something cool about the number 948 that I learned from OEIS.org:

948 and 949 make a Ruth-Aaron pair because they are consecutive numbers and the sum of 948’s prime factors equals the sum of 949’s prime factors:

  • 948 is a composite number.
  • Prime factorization: 948 = 2 × 2 × 3 × 79, which can be written 948 = 2² × 3 × 79
  • 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 948 has exactly 12 factors.
  • Factors of 948: 1, 2, 3, 4, 6, 12, 79, 158, 237, 316, 474, 948
  • Factor pairs: 948 = 1 × 948, 2 × 474, 3 × 316, 4 × 237, 6 × 158, or 12 × 79,
  • Taking the factor pair with the largest square number factor, we get √948 = (√4)(√237) = 2√237 ≈ 30.7896

947 and Level 4

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If you know how to multiply and divide, then you can solve this puzzle. Just use logic to find the factors from 1 to 12 that go in the first column and the top row. Go ahead give it a try!

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

Now here’s a little about the number 947:

947 is a prime number that can be written as the sum of seven consecutive prime numbers:
113 + 127 + 131 + 137 + 139 + 149 + 151 = 947

947 is a palindrome in three other bases:
3B3 BASE 16 (B is 11 base 10), because 3(16²) + 11(16¹) + 3(16⁰) = 947
232 BASE 21 because 2(21²) + 3(21¹) + 2(21⁰) = 947
1L1 BASE 22 (L is 21 BASE 10), because 1(22²) + 21(22¹) + 1(22⁰) = 947

  • 947 is a prime number.
  • Prime factorization: 947 is prime.
  • The exponent of prime number 947 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 947 has exactly 2 factors.
  • Factors of 947: 1, 947
  • Factor pairs: 947 = 1 × 947
  • 947 has no square factors that allow its square root to be simplified. √947 ≈ 30.773365

How do we know that 947 is a prime number? If 947 were not a prime number, then it would be divisible by at least one prime number less than or equal to √947 ≈ 30.8. Since 947 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 947 is a prime number.

What Kind of Shape Is 946 In?

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First of all, 946 is the sum of the numbers from 1 to 43, so it is the 43rd triangular number.

Every other triangular number is also a hexagonal number. Since 946 is the 43rd triangular number, and 43 is an odd number, 946 is also the 22nd hexagonal number. 946 is the 22nd hexagonal number because 22(2(22) – 1) = 22(43) = 946.

But that’s not all. 946 is different than any previous hexagonal number. 946 is the smallest hexagonal number that is also a hexagonal pyramidal number. It is, in fact, the 11th hexagonal pyramidal number. That means if you stack the hexagons in the graphic below in order from largest to smallest, you would get a hexagonal pyramid made with 946 tiny squares. That’s pretty cool, I think.

 

467 + 479 = 946 so 946 is the sum of two consecutive prime numbers.
946 is also the sum of the twenty prime numbers from 11 to 89.

946 is palindrome 181 in BASE 27 because
1(27²) + 8(27¹) + 1(27⁰) = 729 + 216 + 1 = 946

  • 946 is a composite number.
  • Prime factorization: 946 = 2 × 11 × 43
  • 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 946 has exactly 8 factors.
  • Factors of 946: 1, 2, 11, 22, 43, 86, 473, 946
  • Factor pairs: 946 = 1 × 946, 2 × 473, 11 × 86, or 22 × 43
  • 946 has no square factors that allow its square root to be simplified. √946 ≈ 30.75711

There’s Something Odd about the Number 945

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945 = 1 × 3 × 5 × 7 × 9

The sum of the proper divisors of a number determines if the number is abundant, deficient, or perfect. If the sum is greater than the number, the number is abundant. If the sum is less than the number, the number is deficient. If the sum is equal to the number, the number is perfect.

What is a proper divisor? All the factors of a number except itself. Proper divisors are ALMOST the same thing as proper factors. (The number 1 is always a proper divisor, but NEVER a proper factor.)

The first 25 abundant numbers are 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, and 108. Notice that all those numbers are even.

OEIS informs us that 945 is the 232nd abundant number. The first 231 abundant numbers are all even numbers.

Wow, 945 is the smallest ODD abundant number. OEIS also lists the first 31 odd abundant numbers. Every one of the first 31 is divisible by 3 and ends with a 5, but if you scroll down the page you’ll see some that aren’t divisible by 3 or aren’t divisible by 5.

Since 1 × 3 × 5 × 7 × 9 = 945 is the smallest number on the list, you may be wondering about some other numbers:
1 × 3 × 5 × 7 × 9 × 11 = 10,395 made the list.
1 × 3 × 5 × 7 × 9 × 11 × 13 = 135,135 which is too big to be one of the first 31 odd abundant numbers. I was curious if it is also an abundant number, so I found its proper divisors and added them up:

945 is also the hypotenuse of a Pythagorean triple:
567-756-945 which is (3-4-5) times 189

945 looks interesting in a few other bases:
1661 in BASE 8 because 1(8³) + 6(8²) + 6(8¹) + 1(8⁰) = 945
RR in BASE 34 (R is 27 base 10), because 27(34¹) + 27(34⁰) = 27(35) = 945
R0 in BASE 35 because 27(35) + 0(1) = 945

  • 945 is a composite number.
  • Prime factorization: 945 = 3 × 3 × 3 × 5 × 7, which can be written 945 = 3³ × 5 × 7
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 945 has exactly 16 factors.
  • Factors of 945: 1, 3, 5, 7, 9, 15, 21, 27, 35, 45, 63, 105, 135, 189, 315, 945
  • Factor pairs: 945 = 1 × 945, 3 × 315, 5 × 189, 7 × 135, 9 × 105, 15 × 63, 21 × 45, or 27 × 35
  • Taking the factor pair with the largest square number factor, we get √945 = (√9)(√105) = 3√105 ≈ 30.74085