Level 3 Puzzle

1065 and Level 3

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24 and 14 have two common factors, but just one of them will put only numbers from 1 to 12 in the first column. After you write those factors on the puzzle, work down the first column of the puzzle cell by cell writing the appropriate factors as you go.

Print the puzzles or type the solution in this excel file: 12 factors 1063-1072

Here are a few facts about the number 1065:

  • 1065 is a composite number.
  • Prime factorization: 1065 = 3 × 5 × 71
  • 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 1065 has exactly 8 factors.
  • Factors of 1065: 1, 3, 5, 15, 71, 213, 355, 1065
  • Factor pairs: 1065 = 1 × 1065, 3 × 355, 5 × 213, or 15 × 71
  • 1065 has no square factors that allow its square root to be simplified. √1065 ≈ 32.63434

1065 is the hypotenuse of a Pythagorean triple:
639-852-1065 which is (3-4-5) times 213

1065 is a palindrome in two bases:
It’s 353 in BASE 18 because 3(18²) + 5(18) + 3(1) = 1065
1A1 in BASE 28 (A is 10 base 10) because 28² + 10(28) + 1 = 1065

1058 and Level 3

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You can solve this puzzle! Just start with the easy clues closest to the top and work your way down cell by cell until you have the numbers from 1 to 10 in both the first column and the top row. Have fun!

Print the puzzles or type the solution in this excel file: 10-factors-1054-1062

Now I’ll share some information about the number 1058:

  • 1058 is a composite number.
  • Prime factorization: 1058 = 2 × 23 × 23, which can be written 1058 = 2 × 23²
  • The exponents in the prime factorization are 1 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1) = 2 × 3  = 6. Therefore 1058 has exactly 6 factors.
  • Factors of 1058: 1, 2, 23, 46, 529, 1058
  • Factor pairs: 1058 = 1 × 1058, 2 × 529, or 23 × 46
  • Taking the factor pair with the largest square number factor, we get √1058 = (√529)(√2) = 23√2 ≈ 32.52691

1046 and Level 3

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To solve this Level 3 puzzle start with the clues in the first row, 15 and 5. Put their factors in the first column and top row, then work down the puzzle finding the factors of all of the clues. Every factor you write in the first column or top row must be a number from 1 to 12 and can only be used once in each place.

Now here is a little bit about the number 1046:

  • 1046 is a composite number.
  • Prime factorization: 1046 = 2 × 523
  • 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 1046 has exactly 4 factors.
  • Factors of 1046: 1, 2, 523, 1046
  • Factor pairs: 1046 = 1 × 1046 or 2 × 523
  • 1046 has no square factors that allow its square root to be simplified. √1046 ≈ 32.34192

1046 is also palindrome 626 in BASE 13 because 6(13²) + 2(13) + 6 (1) = 1046

1040 and Level 3

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See clues 63 and 72 near the top of this puzzle? Start there and work down cell by cell to find all the factors that will make this puzzle become a multiplication table. Only write each number from 1 to 10 once in the top row and once in the first column.

Print the puzzles or type the solution in this excel file: 10-factors-1035-1043

Now here are some facts about the number 1040:

  • 1040 is a composite number.
  • Prime factorization: 1040 = 2 × 2 × 2 × 2 × 5 × 13, which can be written 1040 = 2⁴ × 5 × 13
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 1040 has exactly 20 factors.
  • Factors of 1040: 1, 2, 4, 5, 8, 10, 13, 16, 20, 26, 40, 52, 65, 80, 104, 130, 208, 260, 520, 1040
  • Factor pairs: 1040 = 1 × 1040, 2 × 520, 4 × 260, 5 × 208, 8 × 130, 10 × 104, 13 × 80, 16 × 65, 20 × 52 or 26 × 40
  • Taking the factor pair with the largest square number factor, we get √1040 = (√13)(√65) = 4√65 ≈ 32.24903.

1040 is the sum of the twelve prime numbers from 61 to 109.
It is also the sum of these four prime numbers:
251 + 257 + 263 + 269 = 1040

1040 is the hypotenuse of FOUR different Pythagorean triples:
256-1008-1040 which is 16 times (16-63-65)
400-960-1040 which is (5-12-13) times 80
528-896-1040 which is 16 times (33-56-65)
624-832-1040 which is (3-4-5) times 208

I like the way 1040 looks in a couple of other bases:
It’s 2020 in BASE 8 because 2(8³) + 2(8) = 2(520) = 1040, and
it’s palindrome 3A3 in BASE 17 (A is 10 base 10) because 3(17²) + 10(17) + 3(1) = 1040

1033 and Level 3

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To solve a level 3 puzzle begin with 80, the clue at the very top of the puzzle. Clue 48 goes with it. What are the factor pairs of those numbers in which both factors are between 1 and 12 inclusive? 80 can be 8×10, and 48 can be 4×12 or 6×8. What is the only number that listed for both 80 and 48? Put that number in the top row over the 80. Put the corresponding factors where they go starting at the top of the first column.

Work down that first column cell by cell finding factors and writing them as you go. Three of the factors have been highlighted because you have to at least look at the 55 and the 5 to deal with the 20 in the puzzle. Have fun!

Print the puzzles or type the solution in this excel file: 12 factors 1028-1034

Here are a few facts about the number 1033:

It is a twin prime with 1031.

32² + 3² = 1033, so it is the hypotenuse of a Pythagorean triple:
192-1015-1033 calculated from 2(32)(3), 32² – 3², 32² + 3²

1033 is a palindrome in two other bases:
It’s 616 in BASE 13 because 6(13²) + 1(13) + 6(1) = 1033
1J1 in BASE 24 (J is 19 base 10) because 24² + 19(24) + 1 = 1033

8¹ + 8º + 8³ + 8³ = 1033 Thanks to OEIS.org for that fun fact!

  • 1033 is a prime number.
  • Prime factorization: 1033 is prime.
  • The exponent of prime number 1033 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1033 has exactly 2 factors.
  • Factors of 1033: 1, 1033
  • Factor pairs: 1033 = 1 × 1033
  • 1033 has no square factors that allow its square root to be simplified. √1033 ≈ 32.1403

How do we know that 1033 is a prime number? If 1033 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1033 ≈ 32.1. Since 1033 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1033 is a prime number.

Here’s another way we know that 1033 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 32² + 3² = 1033 with 32 and 3 having no common prime factors, 1033 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1033 ≈ 32.1. Since 1033 is not divisible by 5, 13, 17, or 29, we know that 1033 is a prime number.

 

1015 and Level 3

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If you glance at this puzzle for a few seconds, you may think there are three places in the top row where the number 1 will work, but in actuality, only one of those places will work with all the other clues in the puzzle. This is a level 3 puzzle, so start with the top cell in the first column and work down cell by cell placing factors in both the first column and the top row until the puzzle resembles a multiplication table.

Print the puzzles or type the solution in this excel file: 12 factors 1012-1018

Here are a few facts about the number 1015:

1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² + 10² + 11² + 12² + 13² + 14² = 1015. That makes 1015 the 14th square pyramidal number.

331 + 337 + 347 = 1015. That’s the sum of 3 consecutive prime numbers.

1015 is the hypotenuse of four Pythagorean triples:
119-1008-1015
168-1001-1015
609-812-1015
700-735-1015

1015 looks interesting when it is written is some other bases:
707 in BASE 12
1D1 in BASE 26 (D is 13 base 10)
TT in BASE 34 (T is 29 base 10)
T0 in BASE 35

  • 1015 is a composite number.
  • Prime factorization: 1015 = 5 × 7 × 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 1015 has exactly 8 factors.
  • Factors of 1015: 1, 5, 7, 29, 35, 145, 203, 1015
  • Factor pairs: 1015 = 1 × 1015, 5 × 203, 7 × 145, or 29 × 35
  • 1015 has no square factors that allow its square root to be simplified. √1015 ≈ 31.85906

1004 and Level 3

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Level 3 puzzles are like Level 4 puzzles on training wheels. You start with the clues at the top of the puzzle and then work down clue by clue until you get to the bottom. For this particular puzzle, I’ve marked three of the clues in red. When you consider the factors of the number 8, you will need to look at those other two red clues in order to proceed because they will eliminate some of the possibilities. Have fun!

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

997 and Level 3

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Can you fill in all the cells of this 12 × 12 mixed up multiplication table if all you are given are the clues given in this puzzle? I promise you it can be done. Start with the two clues at the top of the puzzle and work down clue by clue until you have found all the factors. Afterwards, filling in the rest of the multiplication table will be a breeze.

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

Here are a few facts about 997, the largest three-digit prime number:

31² + 6²  = 997
That means 997 is the hypotenuse of a primitive Pythagorean triple:
372-925-997 calculated from 2(31)(6), 31² – 6², 31² + 6²

  • 997 is a prime number.
  • Prime factorization: 997 is prime.
  • The exponent of prime number 997 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 997 has exactly 2 factors.
  • Factors of 997: 1, 997
  • Factor pairs: 997 = 1 × 997
  • 997 has no square factors that allow its square root to be simplified. √997 ≈ 31.5753

How do we know that 997 is a prime number? If 997 were not a prime number, then it would be divisible by at least one prime number less than or equal to √997 ≈ 31.3. Since 997 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 997 is a prime number.

Here’s another way we know that 997 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 31² + 6² = 997 with 31 and 6 having no common prime factors, 997 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √997 ≈ 31.6. Since 997 is not divisible by 5, 13, 17, or 29, we know that 997 is a prime number.

988 Christmas Factor Tree

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To solve a level 3 puzzle, find two clues in a single row or column. They will help you know what factors to put in the top cell of the first column and two other cells in either the first column or top row. You’ll then be able to work your way down the puzzle by finding the factors of each clue in turn in the most ideal order. The clues below a current clue can still affect the logic of that clue, as you should discover with this puzzle.

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

Here’s a little about the number 988:

It is divisible by 4 because 88 is divisible by 4. Here are a couple of its factor trees that are nicely well-balanced:

988 is the sum of these four consecutive prime numbers:
239 + 241 + 251 + 257 = 988

988 is also the hypotenuse of a Pythagorean triple:
380-912-988 which is (5-12-13) times 76

  • 988 is a composite number.
  • Prime factorization: 988 = 2 × 2 × 13 × 19, which can be written 988 = 2² × 13 × 19
  • 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 988 has exactly 12 factors.
  • Factors of 988: 1, 2, 4, 13, 19, 26, 38, 52, 76, 247, 494, 988
  • Factor pairs: 988 = 1 × 988, 2 × 494, 4 × 247, 13 × 76, 19 × 52, or 26 × 38
  • Taking the factor pair with the largest square number factor, we get √988 = (√4)(√247) = 2√247 ≈ 31.432467

Now, what do you imagine is the total number of triangles of any size in the graphic below? You guessed it, 988.

981 Peppermint Sticks

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This time of year you can buy peppermint sticks that don’t just have red stripes, but they might have green ones, too. Today’s puzzle looks like a couple of peppermint sticks. It will be a sweet experience for you to solve it, so be sure to give it a try.

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

Here is some information about the number 981:

30² + 9² = 981 making 981 the hypotenuse of a Pythagorean triple:
540-819-981 which is 9 times (60-91-109) but can also be calculated from
2(30)(9), 30² – 9², 30² + 9²

981 is palindrome 171 in BASE 28 because 1(28²) + 7(28) + 1(1) = 981

  • 981 is a composite number.
  • Prime factorization: 981 = 3 × 3 × 109, which can be written 981 = 3² × 109
  • 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 981 has exactly 6 factors.
  • Factors of 981: 1, 3, 9, 109, 327, 981
  • Factor pairs: 981 = 1 × 981, 3 × 327, or 9 × 109
  • Taking the factor pair with the largest square number factor, we get √981 = (√9)(√109) = 3√109 ≈ 31.3209