# 1097 and Level 3

72 and 27 are mirror images of each other. What is the largest number that will divide evenly into both of them? Put the answer to that question under the x, and you will have completed the first step in solving this multiplication table puzzle.

Print the puzzles or type the solution in this excel file: 12 factors 1095-1101

Here’s a little bit more about the number 1097:

• 1097 is a prime number.
• Prime factorization: 1097 is prime.
• The exponent of prime number 1097 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1097 has exactly 2 factors.
• Factors of 1097: 1, 1097
• Factor pairs: 1097 = 1 × 1097
• 1097 has no square factors that allow its square root to be simplified. √1097 ≈ 33.12099

How do we know that 1097 is a prime number? If 1097 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1097 ≈ 33.1. Since 1097 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1097 is a prime number.

1097 is the final prime number in the prime triplet, 1091-1093-1097.

1097 is the sum of two squares:
29² + 16² = 1097

1097 is the hypotenuse of a primitive Pythagorean triple:
585-928-1097 calculated from 29² – 16², 2(29)(16), 29² + 16²

Here’s another way we know that 1097 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 29² + 16² = 1097 with 29 and 16 having no common prime factors, 1097 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1097 ≈ 33.1. Since 1097 is not divisible by 5, 13, 17, or 29, we know that 1097 is a prime number.

# 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!

Print the puzzles or type the solution in this excel file: 10-factors-1087-1094

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, 11, 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

# 1057 and Level 2

Can you figure out where to put the numbers from 1 to 10 in both the 1st column and the top row so that this puzzle behaves like a multiplication table?

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

Here’s a little bit about the number 1057:

• 1057 is a composite number.
• Prime factorization: 1057 = 7 × 151
• 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 1057 has exactly 4 factors.
• Factors of 1057: 1, 7, 151, 1057
• Factor pairs: 1057 = 1 × 1057 or 7 × 151
• 1057 has no square factors that allow its square root to be simplified. √1057 ≈ 32.5115

# 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, 11, 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.

# 1047 and Level 4

There are a couple of clues in this puzzle that might be a little tricky, but I know you won’t let that stop you from finding its solution. Puzzles are fun, so have fun with this one.

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

What do I know about the number 1047?

• 1047 is a composite number.
• Prime factorization: 1047 = 3 × 349
• 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 1047 has exactly 4 factors.
• Factors of 1047: 1, 3, 349, 1047
• Factor pairs: 1047 = 1 × 1047 or 3 × 349
• 1047 has no square factors that allow its square root to be simplified. √1047 ≈ 32.357379

1047 is the hypotenuse of a Pythagorean triple:
540-897-1047 which is 3 times (180-299-349)

It is also a palindrome in a couple of bases:
It’s 343 in BASE 18 because 3(18²) + 4(18) + 3(1) = 1047, and
2H2 in BASE 19 (H is 17 base 10) because 2(19²) + 17(19) + 2(1) = 1047

# 1046 and Level 3

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

# 860 and Level 6

Print the puzzles or type the solution on this excel file: 10-factors-853-863

860 is the hypotenuse of a Pythagorean triple: 516-688-860, which is (3-4-5) times 172.

860 can be written as the sum of four consecutive prime numbers: 199 + 211 + 223 + 227 = 860

• 860 is a composite number.
• Prime factorization: 860 = 2 × 2 × 5 × 43, which can be written 860 = 2² × 5 × 43
• 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 860 has exactly 12 factors.
• Factors of 860: 1, 2, 4, 5, 10, 20, 43, 86, 172, 215, 430, 860
• Factor pairs: 860 = 1 × 860, 2 × 430, 4 × 215, 5 × 172, 10 × 86, or 20 × 43,
• Taking the factor pair with the largest square number factor, we get √860 = (√4)(√215) = 2√215 ≈ 29.3257566

# 858 and Level 4

There are sixteen numbers less than 1000 that have four different prime factors. 858 is one of them, and it is the ONLY one that is also a palindrome. Thank you, OEIS.org for alerting us to that fact. No smaller palindrome has four different prime factors!

The sixteen products on that chart each have exactly sixteen factors!

Here’s a Find the Factors 1-10 puzzle for you to solve:

Print the puzzles or type the solution on this excel file: 10-factors-853-863

Here’s a little more about the number 858:

858 is the hypotenuse of a Pythagorean triple: 330-792-858

• 858 is a composite number.
• Prime factorization: 858 = 2 × 3 × 11 × 13
• The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 858 has exactly 16 factors.
• Factors of 858: 1, 2, 3, 6, 11, 13, 22, 26, 33, 39, 66, 78, 143, 286, 429, 858
• Factor pairs: 858 = 1 × 858, 2 × 429, 3 × 286, 6 × 143, 11 × 78, 13 × 66, 22 × 39, or 26 × 33
• 858 has no square factors that allow its square root to be simplified. √858 ≈ 29.291637

# 853 You Can Do This Puzzle!

This is a level 1 puzzle that is easier than even most other level 1 puzzles. You can do this puzzle! If mathematics makes you uncomfortable, you can still do this puzzle! Even if math class is your worse nightmare, you can complete this puzzle, and gain a little confidence. Go ahead, give it a try! Figure out where each number from one to ten goes in the top row and also in the first column so that the puzzle turns into a mixed-up multiplication table. It’s easier and far less time consuming than Sudoku. You CAN do this puzzle! Then, after you find all the factors, and are feeling really good about yourself, IF you want, you can fill in all the other cells of this mixed up multiplication table.

Print the puzzles or type the solution on this excel file: 10-factors-853-863

853 is a prime number that leaves a remainder of 1 when divided by 4, so 853 is the hypotenuse of a Pythagorean triple: 205-828-853.

23² + 18² = 853 so 205-828-853 can be calculated from 23² – 18², 2(23)(18), 23² + 18².

• 853 is a prime number.
• Prime factorization: 853 is prime.
• The exponent of prime number 853 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 853 has exactly 2 factors.
• Factors of 853: 1, 853
• Factor pairs: 853 = 1 × 853
• 853 has no square factors that allow its square root to be simplified. √853 ≈ 29.20616

How do we know that 853 is a prime number? If 853 were not a prime number, then it would be divisible by at least one prime number less than or equal to √853 ≈ 29.2. Since 853 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 853 is a prime number.

Here’s another way we know that 853 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 23² + 18² = 853 with 23 and 18 having no common prime factors, 853 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √853 ≈ 29.2. Since 853 is not divisible by 5, 13, 17, or 29, we know that 853 is a prime number.

# 805 and Level 4

23 × 35 = 805 so we shouldn’t be surprised that 805 is palindrome NN in BASE 34. N is the same as 23 in base 10. Thus NN can be derived from 23(34) + 23(1) = 23(34 + 1) = 23 × 35 = 805. NN obviously is divisible by 11 like all 2 digit palindromes are.

Since 23 = 22 + 1, should we expect that 805 is a palindrome in BASE 22? No, and that is for the same reason that not all multiples of 11 are palindromes.

Finding the factors in today’s puzzle shouldn’t be very difficult, but the last few might be trickier than the rest:

Print the puzzles or type the solution on this excel file: 10-factors 801-806

• 805 is a composite number.
• Prime factorization: 805 = 5 x 7 x 23
• 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 x 2 x 2 = 8. Therefore 805 has exactly 8 factors.
• Factors of 805: 1, 5, 7, 23, 35, 115, 161, 805
• Factor pairs: 805 = 1 x 805, 5 x 161, 7 x 115, or 23 x 35
• 805 has no square factors that allow its square root to be simplified. √805 ≈ 28.37252

805 is the hypotenuse of a Pythagorean triple:

• 483-644-805, which is 3-4-5 times 161

805 can be written as the sum of three squares four ways:

• 25² + 12² + 6² = 805
• 24² + 15² + 2² = 805
• 20² + 18² + 9² = 805
• 18² + 16² + 15² = 805