Level 2 Puzzle

1003 and Level 2

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There is only one way to write the numbers 1 to 10 in both the first column and the top row so that you create a multiplication table and the clues in the puzzle belong where they are. Can you find that way? It easy difficult. Give it a try!

Print the puzzles or type the solution in this excel file: 10-factors-1002-1011

1003 is the hypotenuse of a Pythagorean triple:
472∗885∗1003 which is (8-15-17) times 59

1003 is also palindrome 1101011 in BASE 3
because 3⁶ + 3⁵ + 0(3⁴) + 3³ + 0(3²) + 3¹ + 3⁰ = 1003

  • 1003 is a composite number.
  • Prime factorization: 1003 = 17 × 59
  • 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 1003 has exactly 4 factors.
  • Factors of 1003: 1, 17, 59, 1003
  • Factor pairs: 1003 = 1 × 1003 or 17 × 59
  • 1003 has no square factors that allow its square root to be simplified. √1003 ≈ 31.670175

995 and Level 2

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If you know the only number that will divide evenly into 2, 11, 10, 4, 1, and 9, then you can easily solve this level 2 puzzle. There are even elementary aged kids that you know who can solve it. I’m sure that together you can fill in every cell in the entire puzzle. Don’t be afraid. Just do it.

Print the puzzles or type the solution in this excel file: 12 factors 993-1001

Here’s some stuff that you probably didn’t know about the number 995:

995 has only two factor pairs, and they both use the exact same digits (1, 9, 9, 5):
1 × 995 =  199 × 5

It is the hypotenuse of a Pythagorean triple:
597-796-995 which is (3-4-5) times 199

It is palindrome 3E3 in BASE 16 (E is 14 base 10) because 3(16²) + 14(16) + 3(1) = 995

  • 995 is a composite number.
  • Prime factorization: 995 = 5 × 199
  • 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 995 has exactly 4 factors.
  • Factors of 995: 1, 5, 199, 995
  • Factor pairs: 995 = 1 × 995 or 5 × 199
  • 995 has no square factors that allow its square root to be simplified. √995 ≈ 31.54362

 

 

987 Christmas Star

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Today’s puzzle is a lovely Christmas star whose golden beams shine throughout the dark night. Solving this puzzle could also enlighten your mind.

Print the puzzles or type the solution in this excel file: 10-factors-986-992

987 is made from three consecutive numbers so it is divisible by 3.

It is also the sixteen number in the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, . . .

987 is a palindrome when written in base 11 or base 29:
818 in BASE 11 because 8(121) + 1(11) + 8(1) = 987
151 in BASE 29 because 1(29²) + 5(29) + 1(1) = 987

  • 987 is a composite number.
  • Prime factorization: 987 = 3 × 7 × 47
  • 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 987 has exactly 8 factors.
  • Factors of 987: 1, 3, 7, 21, 47, 141, 329, 987
  • Factor pairs: 987 = 1 × 987, 3 × 329, 7 × 141, or 21 × 47
  • 987 has no square factors that allow its square root to be simplified. √987 ≈ 31.416556

 

979 Was virgács in your shoes this morning?

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Today was Saint Nicholas day in much of Europe. Children woke up and looked in their shoes that they had carefully laid out the night before. They love to find their favorite candies letting them know they’ve been good this last year. In Hungary, where everyone’s behavior is considered to be a mixture of both good and bad, children also found some virgács in their shoes, letting them know they were also naughty some of the time. Today’s puzzle looks a little like virgács.

Print the puzzles or type the solution in this excel file: 12 factors 978-985

1⁴ + 2⁴ + 3⁴ + 4⁴ + 5⁴ = 979

979 is the sum of the eleven prime numbers from 67 to 109.

979 is a palindrome in base 10 and in two other bases:
454 in BASE 15
3D3 in BASE 16 (D is 13 base 10)

979 is the hypotenuse of a Pythagorean triple:
429-880-979 which is 11 times (39-80-89)

  • 979 is a composite number.
  • Prime factorization: 979 = 11 × 89
  • 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 979 has exactly 4 factors.
  • Factors of 979: 1, 11, 89, 979
  • Factor pairs: 979 = 1 × 979 or 11 × 89
  • 979 has no square factors that allow its square root to be simplified. √979 ≈ 31.288976

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.

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

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.

 

 

 

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

934 and Level 2

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I hope you are having a happy Halloween at your house. This puzzle is only a level 2. I’m confident that you can find where all the factors from 1 to 10 go in both the first column and the top row.

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

Now let me tell you a little about the number 934:

934 is palindrome 424 in BASE 15 because
4(15²) + 2(15¹) + 4(15º) = 934

  • 934 is a composite number.
  • Prime factorization: 934 = 2 × 467
  • 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 934 has exactly 4 factors.
  • Factors of 934: 1, 2, 467, 934
  • Factor pairs: 934 = 1 × 934 or 2 × 467
  • 934 has no square factors that allow its square root to be simplified. √934 ≈ 30.56141

Here are some mathematical posts I saw on Twitter to celebrate Halloween:

https://platform.twitter.com/widgets.js

Halloween wishes to all mathematical types out there 😱 pic.twitter.com/0Jyk9IBNmp

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js

926 Creepy Crawler

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You may see lots of crazy creepy-crawling critters this Halloween. Some of them may look very scary.

This level 2 puzzle is really quite tame. Don’t be afraid of it!

Print the puzzles or type the solution on this excel file: 12 factors 923-931

926 is the sum of six consecutive prime numbers:
139 + 149 + 151 + 157 + 163 + 167 = 926

926 is repdigit 222 in BASE 21 because 2(21²) + 2(21) + 2(1) = 2(463) = 926

926 is also palindrome 1C1 in BASE 25 (C is 12 in base 10)

  • 926 is a composite number.
  • Prime factorization: 926 = 2 × 463
  • 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 926 has exactly 4 factors.
  • Factors of 926: 1, 2, 463, 926
  • Factor pairs: 926 = 1 × 926 or 2 × 463
  • 926 has no square factors that allow its square root to be simplified. √926 ≈ 30.430248