1139 and Level 4

The common factors of 30 and 48 are 1, 2, 3, and 6. The rules of a Find the Factors 1 – 12 puzzle require that only numbers from 1 to 12 go in either the first column or the top row. Each number can be used only once in each place. Which common factor of 30 and 48 must be chosen?

Here are some facts about the number 1139:

  • 1139 is a composite number.
  • Prime factorization: 1139 = 17 × 67
  • 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 1139 has exactly 4 factors.
  • Factors of 1139: 1, 17, 67, 1139
  • Factor pairs: 1139 = 1 × 1139 or 17 × 67
  • 1139 has no square factors that allow its square root to be simplified. √1139 ≈ 33.74907

1139 is the sum of seven/eleven consecutive primes:
149 + 151 + 157 + 163 + 167 + 173 + 179 = 1139
79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 = 1139

1139 is the hypotenuse of a Pythagorean triple:
536-1005-1139 which is (8-15-17) times 67

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1126 and Level 4

Since we are only using factors from 1 to 10, we have only one common factor of 12 and 9 to consider instead of two. Also, you will need to ask yourself, “Where is the only place ____ can fit in the first column (or the top row).” to solve this puzzle. Good Luck!

Here are some facts about the number 1126:

  • 1126 is a composite number.
  • Prime factorization: 1126 = 2 × 563
  • 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 1126 has exactly 4 factors.
  • Factors of 1126: 1, 2, 563, 1126
  • Factor pairs: 1126 = 1 × 1126 or 2 × 563
  • 1126 has no square factors that allow its square root to be simplified. √1126 ≈ 33.55592

I chuckled when I noticed that the first five digits of √1126 are all the factors in 1125’s prime factorization.

1126 is palindrome 1K1 in BASE 25 (K is 20 base 10) because 25² + 20(25) + 1 = 1126

1116 Can I Trick You With This Puzzle?

This is only a Level 4 puzzle, but can I trick you into making some assumptions about the puzzle’s factors? I bet I will trick a least a few people. Will you be one of them? If you don’t write a factor unless you are 100% sure it goes where you are putting it, I shouldn’t be able to trick you. Good luck with this one!

Print the puzzles or type the solution in this excel file: 12 factors 1111-1119

Now I’ll share some facts about the number 1116:

  • 1116 is a composite number.
  • Prime factorization: 1116 = 2 × 2 × 3 × 3 × 31, which can be written 1116 = 2² × 3² × 31
  • The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18. Therefore 1116 has exactly 18 factors.
  • Factors of 1116: 1, 2, 3, 4, 6, 9, 12, 18, 31, 36, 62, 93, 124, 186, 279, 372, 558, 1116
  • Factor pairs: 1116 = 1 × 1116, 2 × 558, 3 × 372, 4 × 279, 6 × 186, 9 × 124, 12 × 93, 18 × 62 or 31 × 36
  • Taking the factor pair with the largest square number factor, we get √1116 = (√36)(√31) = 6√31 ≈ 33.40659

1116 looks interesting when it is represented in these other bases:
It’s 13431 in BASE 5 because 1(5⁴) + 3(5³) + 4(5²) + 3(5) + 1(1) = 1116,
VV in BASE 35 (V is 31 base 10) because 31(35) + 31(1) = 31(36) = 1116, and
V0 in BASE 36 because 31(36) = 1116

 

1106 and Level 4

Can you use logic to figure out where all the numbers from 1 to 10 need to go in both the first column and the top row so that this puzzle can become a multiplication table? Give it a try. It’s fun!

Print the puzzles or type the solution in this excel file: 10-factors-1102-1110

Here is some information about the number 1106:

  • 1106 is a composite number.
  • Prime factorization: 1106 = 2 × 7 × 79
  • 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 1106 has exactly 8 factors.
  • Factors of 1106: 1, 2, 7, 14, 79, 158, 553, 1106
  • Factor pairs: 1106 = 1 × 1106, 2 × 553, 7 × 158, or 14 × 79
  • 1106 has no square factors that allow its square root to be simplified. √1106 ≈ 33.25658

1106 is repdigit 222 in BASE 23 because 2(23² + 23+ 1) = 2(553) = 1106

1098 a Lucky Level 4 Puzzle?

You are lucky that this puzzle has some easy clues in it. You will have no problem getting started. Be warned, later on, you may not feel so lucky! I’m sure you can solve it if you keep with it.

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

Now, here are a few facts about the number 1098:

  • 1098 is a composite number.
  • Prime factorization: 1098 = 2 × 3 × 3 × 61, which can be written 1098 = 2 × 3² × 61
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1098 has exactly 12 factors.
  • Factors of 1098: 1, 2, 3, 6, 9, 18, 61, 122, 183, 366, 549, 1098
  • Factor pairs: 1098 = 1 × 1098, 2 × 549, 3 × 366, 6 × 183, 9 × 122, or 18 × 61,
  • Taking the factor pair with the largest square number factor, we get √1098 = (√9)(√122) = 3√122 ≈ 33.13608

This first fact from Stetson.edu uses only digits found in 1098. It makes 1098 look pretty lucky:
1098 = 11 + 0 + 999 + 88

1098 is the sum of four consecutive prime numbers:
269 + 271 + 277 + 281 = 1098

1098 is the hypotenuse of a Pythagorean triple:
198-1080-1098 which is 18 times (11-60-61)

1098 is a palindrome when it is written in three different bases:
It’s 2112 in BASE 8 because 2(8³) + 8² + 8 + 2(1) = 1098,
909 in BASE 11 because 9(11² + 1) = 9(122) = 1098, and it’s
666 in BASE 13. Oh my! How unlucky can you get? Why does it have two unlucky numbers, 666 and 13? Because 6(13² + 13 + 1) = 6(183) = 1098.

 

1090 and Level 4

This puzzle has both 54 and 56 as clues. Many people get the factors involved (6, 7, 8, 9) mixed up. Remember these two multiplication facts:
6 7 × 8 9 = 54
6 7 × 8 9 = 56
The closer factors make the bigger number.

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

Now I’ll share some information about the number 1090:

  • 1090 is a composite number.
  • Prime factorization: 1090 = 2 × 5 × 109
  • 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 1090 has exactly 8 factors.
  • Factors of 1090: 1, 2, 5, 10, 109, 218, 545, 1090
  • Factor pairs: 1090 = 1 × 1090, 2 × 545, 5 × 218, or 10 × 109
  • 1090 has no square factors that allow its square root to be simplified. √1090 ≈ 33.01515

1090 is the sum of the fourteen prime numbers from 47 to 107.

1090 is the sum of two squares two different ways:
27² + 19² = 1090
33² +  1² = 1090

1090 is the hypotenuse of four Pythagorean triples:
66-1088-1090 calculated from 2(33)(1), 33² –  1², 33² +  1²;
it is also 2 times (33-544-545),
368-1026-1090 calculated from 27² – 19² , 2(27)(19) , 27² + 19²;
it is also 2 times (184-513-545),
600-910-1090 which is 10 times (60-91-109)
654-872-1090 which is (3-4-5) times 218

1090 is a palindrome in two different bases:
It’s 1441 in BASE 9 because 1(9³) + 4(9²) + 4(9) + 1(1) = 1090
101 in BASE 33 because 33² + 1 = 1090

1076 and Level 4

 

If you roll a pair of dice, you are taking a chance that the roll might not be favorable for you. Don’t take your chances when solving this puzzle. It can be solved completely by relying on logic. It won’t require any luck, but “Good luck!” anyway.

Print the puzzles or type the solution in this excel file: 10-factors-1073-1079

Here is some information about the number 1076:

  • 1076 is a composite number.
  • Prime factorization: 1076 = 2 × 2 × 269, which can be written 1076 = 2² × 269
  • 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 1076 has exactly 6 factors.
  • Factors of 1076: 1, 2, 4, 269, 538, 1076
  • Factor pairs: 1076 = 1 × 1076, 2 × 538, or 4 × 269
  • Taking the factor pair with the largest square number factor, we get √1076 = (√4)(√269) = 2√269 ≈ 32.80244

1076 is the hypotenuse of a Pythagorean triple:
276-1040-1076 which is 4 times (69-260-269)

1076 is a palindrome in a couple of bases:
It’s 434 in BASE 16 because 4(16²) + 3(16) +4(1) = 1076, and
it’s 1I1 in BASE 25 (I is 18 base 10) because 25² + 18(25) + 1 = 1076

1066 and Level 4

Some of the clues in today’s puzzle were used in previous puzzles this week. Sometimes their factors have to be exactly the same as they were in the previous puzzles, but sometimes they might not be. Can you figure out where to put the numbers 1 to 12 in both the first column and the top row so that those numbers are the factors of the clues given?

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

Here are a few facts about the number 1066:

  • 1066 is a composite number.
  • Prime factorization: 1066 = 2 × 13 × 41
  • 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 1066 has exactly 8 factors.
  • Factors of 1066: 1, 2, 13, 26, 41, 82, 533, 1066
  • Factor pairs: 1066 = 1 × 1066, 2 × 533, 13 × 82, or 26 × 41
  • 1066 has no square factors that allow its square root to be simplified. √1066 ≈ 32.649655

29² + 15² = 1066
25² + 21² = 1066

1066 is the hypotenuse of FOUR Pythagorean triples:
616-870-1066 is 2 times (308-435-533) and calculated from 29² – 15², 2(29)(15), 29² + 15²
410-984-1066 which is (5-12-13) times 82
234-1040-1066 which is 26 times (9-40-41)
184-1050-1066 is 2 times (92-525-533) and calculated from 25² – 21², 2(25)(21), 25² + 21²

1066 looks interesting when it is written in some other bases:
It’s palindrome 1110111 in BASE 3 because 3⁶ + 3⁵ + 3⁴ + 3² + 3¹ +3⁰ = 1066,
13231 in BASE 5 because 1(5⁴) + 3(5³) + 2(5²) + 3(5) + 1(1) = 1066,
1414 in BASE 9 because 1(9³) + 4(9²) + 1(9) + 4(1) = 1066, and
2I2 in BASE 19 (I is 18 base 10) because 2(19²) + 18(19) + 2(1) = 1066

 

 

1059 and Level 4

A level four puzzle is only a little more difficult than a level three puzzle. Instead of starting with the top clue and working down cell by cell, the next clue that you need could be anywhere in the puzzle. It may be a little harder, but you can still solve this puzzle!

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

What can I tell you about the number 1059?

  • 1059 is a composite number.
  • Prime factorization: 1059 = 3 × 353
  • 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 1059 has exactly 4 factors.
  • Factors of 1059: 1, 3, 353, 1059
  • Factor pairs: 1059 = 1 × 1059 or 3 × 353
  • 1059 has no square factors that allow its square root to be simplified. √1059 ≈ 32.54228

1059 is the hypotenuse of a Pythagorean triple:
675-816-1059 which is 3 times (225-272-353)

1059 is palindrome 636 in BASE 13 because 6(13²) + 3(13) + 6(1) = 1059

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