1091 and Level 5

Can you figure out where to put all the numbers from 1 to 10 in both the first column and the top row so that those factors and the clues can become a multiplication table? Some of the clues might be a little tricky, but I’m sure you can figure them all out.

Here are a few facts about the number 1091:

  • 1091 is a prime number.
  • Prime factorization: 1091 is prime.
  • The exponent of prime number 1091 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1091 has exactly 2 factors.
  • Factors of 1091: 1, 1091
  • Factor pairs: 1091 = 1 × 1091
  • 1091 has no square factors that allow its square root to be simplified. √1091 ≈ 33.03029

How do we know that 1091 is a prime number? If 1091 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1091 ≈ 33. Since 1091 cannot be divided evenly by 2, 3, 5, 7, 3, 13, 17, 19, 23, 29 or 31, we know that 1091 is a prime number.

1091 is the first prime number in the prime triplet (1091, 1093, 1097).

1091 looks interesting when it is written in some other bases:
It’s 13331 in BASE 5 because 1(5⁴) + 3(5³) + 3(5²) + 3(5) + 1(1) = 1091,
3D3 in BASE 17 (D is 13 base 10) because 3(17²) + 13(17) + 3(1) = 1091,
and it’s 123 in BASE 32 because 1(32²) + 2(32) + 3(1) = 1091

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1087 and Level 3

Using logic, start with the clue on the top row and work yourself down row by row filling in the appropriate factors while you go. You might find this level 3 puzzle a little tricky near the bottom of the puzzle, so I didn’t want to wait to share it with you. Happy factoring!

1087 is the first prime since 1069, which was 18 numbers ago! What else can I tell you about it?

  • 1087 is a prime number.
  • Prime factorization: 1087 is prime.
  • The exponent of prime number 1087 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1087 has exactly 2 factors.
  • Factors of 1087: 1, 1087
  • Factor pairs: 1087 = 1 × 1087
  • 1087 has no square factors that allow its square root to be simplified. √1087 ≈ 32.96968

How do we know that 1087 is a prime number? If 1087 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1087 ≈ 33. Since 1087 cannot be divided evenly by 2, 3, 5, 7, 3, 13, 17, 19, 23, 29 or 31, we know that 1087 is a prime number.

1087 is also palindrome 767 in BASE 12 because
7(12²) + 6(12) + 7(1) = 1087

1063 and Level 1

Lucky you found this puzzle today! You can solve it by writing the factors 1 to 12 in both the first column and the top row so that the given clues are the products of the corresponding factors.

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

Now I’ll share a little information about the number 1063:

  • 1063 is a prime number. 1061 and 1063 are twin primes.
  • Prime factorization: 1063 is prime.
  • The exponent of prime number 1063 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1063 has exactly 2 factors.
  • Factors of 1063: 1, 1063
  • Factor pairs: 1063 = 1 × 1063
  • 1063 has no square factors that allow its square root to be simplified. √1063 ≈ 32.60368

How do we know that 1063 is a prime number? If 1063 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1063 ≈ 32.6. Since 1063 cannot be divided evenly by 2, 3, 5, 7, 3, 13, 17, 19, 23, 29 or 31, we know that 1063 is a prime number.

1063 is the sum of seven consecutive prime numbers:
137 + 139 + 149 + 151 + 157 + 163 + 167 = 1063

1063 is a high flying palindrome in one other base:
It’s 747 in BASE 12 because 7(12²) + 4(12) + 7(1) = 1063

1061 and Level 5

Study the clues in the puzzle below. If you begin with the right set of clues, the puzzle can be solved quite easily, but if you don’t, you might get tripped up. Good luck!

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

Here are a few facts about the number 1061:

  • 1061 is a prime number. 1061 and 1063 are twin primes.
  • Prime factorization: 1061 is prime.
  • The exponent of prime number 1061 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1061 has exactly 2 factors.
  • Factors of 1061: 1, 1061
  • Factor pairs: 1061 = 1 × 1061
  • 1061 has no square factors that allow its square root to be simplified. √1061 ≈ 32.57299

How do we know that 1061 is a prime number? If 1061 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1061 ≈ 32.6. Since 1061 cannot be divided evenly by 2, 3, 5, 7, 3, 13, 17, 19, 23, 29 or 31, we know that 1061 is a prime number.

1061 is the sum of the 17 prime numbers from 29 to 101, and it also is the sum of these three consecutive prime numbers: 349 + 353 + 359 = 1061

31² + 10² = 1061 so 1061 is the hypotenuse of a Pythagorean triple:
620-861-1061, a primitive calculated from 2(31)(10), 31² – 10², 31² + 10²

Here’s another way we know that 1061 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 31² + 10² = 1061 with 31 and 10 having no common prime factors, 1061 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1061 ≈ 32.6. Since 1061 is not divisible by 5, 13, 17, or 29, we know that 1061 is a prime number.

 

 

 

 

1051 is the 21st Centered Pentagonal Number

1051 is the 21st centered pentagonal number. It is exactly 100 more than the previous centered pentagonal number because there are exactly 100 little blue squares on the outside-most pentagon in the graphic below.

Can you see the five triangles surrounding the center square? Each of them has the same number of tiny squares and indicates that 1051 is 1 more than five times the 20th triangular number:
1 + 5(20)(21)/2 = 1 + 50(21) = 1051

1049 and 1051 are twin primes.

  • 1051 is a prime number.
  • Prime factorization: 1051 is prime.
  • The exponent of prime number 1051 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1051 has exactly 2 factors.
  • Factors of 1051: 1, 1051
  • Factor pairs: 1051 = 1 × 1051
  • 1051 has no square factors that allow its square root to be simplified. √1051 ≈ 32.41913

How do we know that 1051 is a prime number? If 1051 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1051 ≈ 32.4. Since 1051 cannot be divided evenly by 2, 3, 5, 7, 3, 13, 17, 19, 23, 29 or 31, we know that 1051 is a prime number.

1051 is a palindrome when it is written in three other bases:
It’s 737 in BASE 12 because 7(144) + 3(12) + 7(1) = 1051,
1H1 in BASE 25 (H is 17 base 10) because 25² +17(25) + 1 = 1051, and
151 in BASE 30 because 30² + 5(30) + 1 = 1051

1049 and Level 6

Find the Factors Puzzles are always solved using logic. Can you see the logic needed to solve this one?

Print the puzzles or type the solution in this excel file: 12 factors 1044-1053

Here are a few facts about the number 1049:

  • 1049 is a prime number.
  • Prime factorization: 1049 is prime.
  • The exponent of prime number 1049 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1049 has exactly 2 factors.
  • Factors of 1049: 1, 1049
  • Factor pairs: 1049 = 1 × 1049
  • 1049 has no square factors that allow its square root to be simplified. √1049 ≈ 32.38827

How do we know that 1049 is a prime number? If 1049 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1049 ≈ 32.4. Since 1049 cannot be divided evenly by 2, 3, 5, 7, 3, 13, 17, 19, 23, 29 or 31, we know that 1049 is a prime number.

1049 is also the sum of three consecutive prime numbers:
347 + 349 + 353 = 1049

32² + 5² = 1049 so 1049 is the hypotenuse of a Pythagorean triple:
320-999-1049 calculated from 2(32)(5), 32² – 5², 32² + 5²

Here’s another way we know that 1049 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 32² + 5² = 1049 with 32 and 5 having no common prime factors, 1049 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1049 ≈ 32.4. Since 1049 is not divisible by 5, 13, 17, or 29, we know that 1049 is a prime number.

 

 

1039 and Level 2

The eleven clues in this puzzle are enough to figure out where to place the factors and then complete the entire multiplication table. Try it. I know you can solve it!

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

Now here’s a little about the number 1039:

  • 1039 is a prime number.
  • Prime factorization: 1039 is prime.
  • The exponent of prime number 1039 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1039 has exactly 2 factors.
  • Factors of 1039: 1, 1039
  • Factor pairs: 1039 = 1 × 1039
  • 1039 has no square factors that allow its square root to be simplified. √1039 ≈ 32.23352

How do we know that 1039 is a prime number? If 1039 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1039 ≈ 32.2. Since 1039 cannot be divided evenly by 2, 3, 5, 7, 3, 13, 17, 19, 23, 29 or 31, we know that 1039 is a prime number.

1039 is a palindrome when it is written in two different bases:
It’s 727 in BASE 12 because 7(144) + 2(12) + 7(1) = 1039, and
494 in BASE 15 because 4(225) + 9(15) + 4(1) = 1039

1033 and Level 3

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 Stetson.edu 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, 3, 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.

 

1031 Prepare for World Maths Day 2018

You might think that a day lasts 24 hours, but strategic use of the international date line can actually make a single day last 48 hours!

How will you spend the 48 hour day that will be 7 March 2018?

Colleen Young encourages you and your class to register for and participate in World Maths Day 2018 held that day. She shares the necessary links as well as several tips on how to prepare.

One way to prepare now is playing multiplication games like the level 1 puzzle below. The puzzle is just a multiplication table but the factors are missing and only a few of the products are given, and they aren’t in the order you would normally expect. Can you figure out where the factors from 1 to 12 belong in both the first column and the top row of the puzzle?

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

If this puzzle is too easy for you, Then it is time to move on to a level 2 or higher puzzle. You can find one in the link above and plenty others here at findthefactors.com.

Now I’d like to tell you some things that I’ve learned about the number 1031:

1031 and 1033 are twin primes.

1031 is a palindrome in a couple of bases:
It’s 858 in BASE 11 because 8(121) + 5(11) + 8(1) = 1031 and
it’s 272 in BASE 21 because 2(441) + 7(21) + 2(1) = 1031

  • 1031 is a prime number.
  • Prime factorization: 1031 is prime.
  • The exponent of prime number 1031 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1031 has exactly 2 factors.
  • Factors of 1031: 1, 1031
  • Factor pairs: 1031 = 1 × 1031
  • 1031 has no square factors that allow its square root to be simplified. √1031 ≈ 32.10919

How do we know that 1031 is a prime number? If 1031 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1031 ≈ 32.1. Since 1031 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1031 is a prime number.

1021 Mystery Level

Here is the second puzzle in a week’s worth of mystery level puzzles. Will it be very difficult or not so bad? That’s the mystery. You’ll have to try it to know for sure. You only need to use logic and your knowledge of the multiplication table to solve it. I wish you luck!

Print the puzzles or type the solution in this excel file: 10-factors-1019-1027

That puzzle may have been a mystery, but the number 1021 isn’t much of a mystery at all. It is the second number of twin primes, 1019 and 1021.

Since it is a prime number, and it has a remainder of one when it is divided by 4, it can be written as the sum of two squares:
30² + 11² = 1021.

Since it can be written as the sum of two squares, it is the hypotenuse of a Pythagorean triple:
660-779-1021 calculated from 2(30)(11), 30² – 11², 30² + 11²

It is also palindrome 141 in BASE 30 because 1(30²) + 4(30) + 1(1) = 1021

  • 1021 is a prime number.
  • Prime factorization: 1021 is prime.
  • The exponent of prime number 1021 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1021 has exactly 2 factors.
  • Factors of 1021: 1, 1021
  • Factor pairs: 1021 = 1 × 1021
  • 1021 has no square factors that allow its square root to be simplified. √1021 ≈ 31.95309

How do we know that 1021 is a prime number? If 1021 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1021 ≈ 31.95. Since 1021 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1021 is a prime number.

Here’s another way we know that 1021 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 30² + 11² = 1021 with 30 and 11 having no common prime factors, 1021 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1021 ≈ 31.95. Since 1021 is not divisible by 5, 13, 17, or 29, we know that 1021 is a prime number.