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

1002 Merry Christmas!

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Merry Christmas, everyone!

No matter what is going on in your life, may your day today be filled with love, peace, and joy.

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

1002 is the sum of the eighteen prime numbers from 19 to 97.
It is also the sum of two consecutive prime numbers:
499 + 503 = 1002

1002 is a palindrome in two other bases:
6B6 in BASE 12 (B is 11 base 10)
2A2 in BASE 20 (A is 10 base 10)

  • 1002 is a composite number.
  • Prime factorization: 1002 = 2 × 3 × 167
  • 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 1002 has exactly 8 factors.
  • Factors of 1002: 1, 2, 3, 6, 167, 334, 501, 1002
  • Factor pairs: 1002 = 1 × 1002, 2 × 501, 3 × 334, or 6 × 167
  • 1002 has no square factors that allow its square root to be simplified. √1002 ≈ 31.65438

1001 and Level 6

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There is only one multiplication table that has the numbers you see in this puzzle exactly where you see them here. Can you find the factors that create that multiplication table? This is a level 6 puzzle so it won’t be easy, but it can be done by just using logic and the basic 1-12 multiplication facts.

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

1001 is the product of three consecutive prime numbers:
7 × 11 × 13 = 1001

1001 is also the sum of fifteen consecutive prime numbers:
37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 = 1001

1001 is the 26th pentagonal number.

1001 is the hypotenuse of a Pythagorean triple:
385² + 924² = 1001²

1001 is a palindrome in base 10 and in base 25:
It’s 1F1 in BASE 25 (F is 15 base 10) because 1(625) + 15(25) + 1(1) = 1001

  • 1001 is a composite number.
  • Prime factorization: 1001 = 7 × 11 × 13
  • 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 1001 has exactly 8 factors.
  • Factors of 1001: 1, 7, 11, 13, 77, 91, 143, 1001
  • Factor pairs: 1001 = 1 × 1001, 7 × 143, 11 × 91, or 13 × 77
  • 1001 has no square factors that allow its square root to be simplified. √1001 ≈ 31.63858

1000 Find the Factors Challenge

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Dan Bach asked me if I had anything special planned for my 1000 post. I had a few things planned, but this Find the Factors Challenge exists because of his question and our twitter conversation. I like this puzzle better than my previous plans. It is only fitting that I dedicate this puzzle to Dan Bach.

There are four places outlined in bold that need you to place the numbers 1 – 10 so that each section of the puzzle will work as a multiplication table. Each section of the puzzle is dependent on the other sections. None of the sections could be a puzzle independent of the others. You will use similar logic to solve this puzzle as you would a regular Find the Factors puzzle. It will not be easy to solve at all, but give it a try. There is only one solution.

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

1000 is a perfect cube as well as the product of perfect cubes.

30² + 10² = 1000
26² + 18² = 1000

1000 is the hypotenuse of three Pythagorean triples:
600² + 800² = 1000²
352² + 936² = 1000²
280² + 960² = 1000²

1000 looks interesting in a few bases:
2626 in BASE 7
1331 in BASE 9
1A1 in BASE 27 (A is 10 base 10)

  • 1000 is a composite number.
  • Prime factorization: 1000 = 2 × 2 × 2 × 5 × 5 × 5, which can be written 1000 = 2³ × 5³
  • The exponents in the prime factorization are 3 and 3. Adding one to each and multiplying we get (3 + 1)(3 + 1) = 4 × 4 = 16. Therefore 1000 has exactly 16 factors.
  • Factors of 1000: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000
  • Factor pairs: 1000 = 1 × 1000, 2 × 500, 4 × 250, 5 × 200, 8 × 125, 10 × 100, 20 × 50, or 25 × 40
  • Taking the factor pair with the largest square number factor, we get √1000 = (√100)(√10) = 10√10 ≈ 31.622777

999 and Level 5

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What are the common factors of 6 and 12? There are many: 1, 2, 3, and 6 are all common factors and possibilities for this puzzle. Only one of those numbers will work with the other clues to make this puzzle a multiplication table. That why 6 and 12 is NOT a good place to start when solving this puzzle. Look at all the other rows and columns with two or more clues. ONE of them will be the perfect place to start. Good luck!

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

999 is the smallest number whose digits add up to 27.

999 is 666 upside-down.

Did you know that 999² = 998001? That’s cool because 998 + 001 = 999. Numbers whose squares can be separated and then added together to get the original number are so unusual, they’ve been given a name, Kaprekar numbers. There are only seven Kaprekar numbers less than 999. OEIS.org was my source for this fun 999 fact.

999 is the hypotenuse of a Pythagorean triple:
324-945-999 which is 27 times (12-35-37)

999 is a repdigit in base 10 and it looks interesting in some other bases, too:
It’s 4343 in BASE 6 because 4(6³) + 3(6²) + 4(6¹) + 3(6⁰) = 999,
It’s palindrome 515 in BASE 14 because 5(14²) + 1(14) + 5(1) = 999,
and it’s repdigit RR in BASE 36 (R is 27 base 10) because 27(36) + 27(1) = 27(37) = 999

  • 999 is a composite number.
  • Prime factorization: 999 = 3 × 3 × 3 × 37, which can be written 999 = 3³ × 37
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 999 has exactly 8 factors.
  • Factors of 999: 1, 3, 9, 27, 37, 111, 333, 999
  • Factor pairs: 999 = 1 × 999, 3 × 333, 9 × 111, or 27 × 37
  • Taking the factor pair with the largest square number factor, we get √999 = (√9)(√111) = 3√111 ≈ 31.60696

998 and Level 4

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I could easily turn this level 4 puzzle into a level 3 puzzle by rearranging the clues so that you would know an ideal order to work on the clues. Since I’m not doing that, you will have to think more to solve this or any other level 4 puzzle. It still won’t be that hard to do. You will just have to look at all the clues and think about which one should be used next. Remember to always use logic when you consider each clue. Guessing and checking will only frustrate you!

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

Here’s a little bit about the number 998:

998 is palindrome 828 in BASE 11 because 8(121) + 2(11) + 8(1) = 998

  • 998 is a composite number.
  • Prime factorization: 998 = 2 × 499
  • 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 998 has exactly 4 factors.
  • Factors of 998: 1, 2, 499, 998
  • Factor pairs: 998 = 1 × 998 or 2 × 499
  • 998 has no square factors that allow its square root to be simplified. √998 ≈ 31.5911

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.

996 Christmas Factor Tree

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The puzzles this week might look rather plain, but together the seven puzzles make a lovely Christmas tree factoring puzzle. The difficulty level of each of the puzzles is not identified. Some of them are very easy, and some of them are difficult. Some are in-between. How many of them can you solve?

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

Here’s a little about the number 996:

Here are a few of its possible factor trees. They look a little like Christmas trees, too.

 

Usually,  I only go up to base 36 when I look for palindromes or repdigits. 966 is NOT a palindrome or repdigit in any of those bases, but can it ever be one? To me, repdigits are more interesting than palindromes because you can find them by factoring.  966 has 6 factors greater than 36: 83, 166, 249, 332, 498, 996. If you subtract 1 from each of those, then 996 will be a repdigit in each of those bases.
In BASE 82, it’s CC (C is 12 base 10)
In BASE 165, it’s 66
In BASE 248, it’s 44
In BASE 331, it’s 33
In BASE 497, it’s 22
In BASE 995, it’s 11
Don’t be surprised when I tell you that 12, 6, 4, 3, 2, and 1 are also factors of 996!

  • 996 is a composite number.
  • Prime factorization: 996 = 2 × 2 × 3 × 83, which can be written 996 = 2² × 3 × 83
  • 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 996 has exactly 12 factors.
  • Factors of 996: 1, 2, 3, 4, 6, 12, 83, 166, 249, 332, 498, 996
  • Factor pairs: 996 = 1 × 996, 2 × 498, 3 × 332, 4 × 249, 6 × 166, or 12 × 83,
  • Taking the factor pair with the largest square number factor, we get √996 = (√4)(√249) = 2√249 ≈ 31.55947

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

 

 

994 and Level 1

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All you need is these eleven clues and the multiplication facts in a normal 12 × 12 multiplication table to completely fill in every square of this abnormal multiplication table, I mean puzzle. Don’t worry about how fast you can solve the puzzle. The more puzzles you solve the better you will get at doing them. Relax and enjoy yourself!

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

When is the number 994 a palindrome?
It is 4334 in BASE 6 because 4(6³) + 3(6²) + 3(6¹) + 4(6⁰) = 994, and
it’s 464 in BASE 15 because 4(15²) + 6(15¹) + 4(15⁰) = 994

  • 994 is a composite number.
  • Prime factorization: 994 = 2 × 7 × 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 994 has exactly 8 factors.
  • Factors of 994: 1, 2, 7, 14, 71, 142, 497, 994
  • Factor pairs: 994 = 1 × 994, 2 × 497, 7 × 142, or 14 × 71
  • 994 has no square factors that allow its square root to be simplified. √994 ≈ 31.52777