1044 and Level 1

All of the clues in this puzzle have three common factors, but only one of those three factors won’t put a number greater than twelve in either the first column or the top row. Can you figure out what that common factor is as well as all the other factors that belong in this puzzle?

Now I’ll share some information about the number 1044:

  • 1044 is a composite number.
  • Prime factorization: 1044 = 2 × 2 × 3 × 3 × 29, which can be written 1044 = 2² × 3² × 29
  • 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 1044 has exactly 18 factors.
  • Factors of 1044: 1, 2, 3, 4, 6, 9, 12, 18, 29, 36, 58, 87, 116, 174, 261, 348, 522, 1044
  • Factor pairs: 1044 = 1 × 1044, 2 × 522, 3 × 348, 4 × 261, 6 × 174, 9 × 116, 12 × 87, 18 × 58 or 29 × 36
  • Taking the factor pair with the largest square number factor, we get √1044 = (√36)(√29) = 6√29 ≈ 32.31099

30² + 12² =1044

1044 is the hypotenuse of a Pythagorean triple:
720-756-1044 calculated from 2(30)(12), 30² – 12², 30² + 12².
It is also (20-21-29) times 36.

1044 is the sum of twin primes: 521 + 523 = 1044

1044 looks interesting a few other bases:
It’s 414 in BASE 16 because 4(16²) + 1(16) + 4(1) = 1044,
TT in BASE 35 (T is 29 base 10) because 29(35) + 29(1) = 29(35 + 1) = 29(36) = 1044, and T0 in BASE 36 because 29(36) = 1044

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1043 Find the Factors Challenge Puzzle

I made this particular Find the Factors 1-10 Challenge puzzle three weeks ago. It took me just under 30 minutes to solve it when I tried it again before publishing it. How long will it take you to 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 1043:

  • 1043 is a composite number.
  • Prime factorization: 1043 = 7 × 149
  • 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 1043 has exactly 4 factors.
  • Factors of 1043: 1, 7, 149, 1043
  • Factor pairs: 1043 = 1 × 1043 or 7 × 149
  • 1043 has no square factors that allow its square root to be simplified. √1043 ≈ 32.29551

1043 is the sum of consecutive prime numbers two different ways:
It’s the sum of the 21 prime numbers from 11 to 97 and,
it’s the sum of the 13 prime numbers from 53 to 107.

1043 is also the hypotenuse of a Pythagorean triple:
357-980-1043 which is 7 times (51-140-149)

 

1042 and Level 6

I’ve already published the two level 5 puzzles that are in this week’s set of puzzles. If going from a level 4 puzzle to a level 6 puzzle is too big of a jump for you, then try either one of those two level 5 puzzles first. You can find them as well as this puzzle in the link below the puzzle.

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

Here are a few facts about the number 1042:

  • 1042 is a composite number.
  • Prime factorization: 1042 = 2 × 521
  • 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 1042 has exactly 4 factors.
  • Factors of 1042: 1, 2, 521, 1042
  • Factor pairs: 1042 = 1 × 1042 or 2 × 521
  • 1042 has no square factors that allow its square root to be simplified. √1042 ≈ 32.28002

31²  + 9² = 1042

1042 is the hypotenuse of a Pythagorean triple:
558-880-1042 calculated from 2(31)(9), 31²  – 9², 31²  + 9²

1042 is also a palindrome in a couple of bases:
It’s 868 in BASE 11 because 8(121) + 6(11) + 8(1) = 1042, and
2C2 in BASE 20 (C is 12 in base 10) because 2(400) + 10(20) + 2(1) = 1042

 

 

 

1041 and Level 4

Any level 3 puzzle can be easily made into a level 4 puzzle by removing some restrictions on the order of the clues. If you can solve a level 3 puzzle, then this level 4 puzzle will be only a little more difficult to solve.

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

What have I found out about the number 1041?

  • 1041 is a composite number.
  • Prime factorization: 1041 = 3 × 347
  • 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 1041 has exactly 4 factors.
  • Factors of 1041: 1, 3, 347, 1041
  • Factor pairs: 1041 = 1 × 1041 or 3 × 347
  • 1041 has no square factors that allow its square root to be simplified. √1041 ≈ 32.26453

1041 is also a palindrome in three other bases:
It’s 13131 in BASE 5 because 5⁴ + 3(5³) + 5² +3(5) + 1 = 1041,
545 in BASE 14 because 5(14²) + 4(14) + 5(1) = 1041, and
1E1 in BASE 26 (E is 14 base 10) because 26² + 14(26) + 1 = 1041

1040 and Level 3

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

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

1038 Hoppy Easter/April Fool’s Day

A trickster Easter bunny left some of my grandchildren candy and other treasures, not in a traditional Easter basket but in pink, green, and orange pumpkins! It’s April Fool’s Day so we shouldn’t be surprised.

That same trickster bunny has a purple pumpkin puzzle for YOU to try today, too. Part of the puzzle is easy while other parts are tricky: Is it 3 or 5 that is the common factor of 30 and 15 that makes this puzzle work?

Hmm…can you figure out where to put the numbers 1 to 10 in the first column and the top row or will you be tricked this April Fool’s Day?

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

Now I’ll tell you a little bit about the number 1038.

1 + 8 is divisible by 3 so 1038 is also divisible by 3. (Including multiples of 3 in the sum isn’t necessary for that divisibility trick to work.)

  • 1038 is a composite number.
  • Prime factorization: 1038 = 2 × 3 × 173
  • 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 1038 has exactly 8 factors.
  • Factors of 1038: 1, 2, 3, 6, 173, 346, 519, 1038
  • Factor pairs: 1038 = 1 × 1038, 2 × 519, 3 × 346, or 6 × 173
  • 1038 has no square factors that allow its square root to be simplified. √1038 ≈ 32.218007

1038 is also the hypotenuse of a Pythagorean triple:
312-990-1038 which is 6 times (52-165-173)

1037 and Level 1

This week I’ll start with this level 1 puzzle that is just about as easy as they get. If you’ve never done a Find the Factors puzzle, you can still easily figure this one out. The finished puzzle will look like a regular multiplication table but with the factors in a different order than usual. Go ahead, give this one a try!

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

Now I’ll write some facts about the number 1037:

  • 1037 is a composite number.
  • Prime factorization: 1037 = 17 × 61
  • 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 1037 has exactly 4 factors.
  • Factors of 1037: 1, 17, 61, 1037
  • Factor pairs: 1037 = 1 × 1037 or 17 × 61
  • 1037 has no square factors that allow its square root to be simplified. √1037 ≈ 32.20248

1037 is palindrome 191 in BASE 28 because 28² + 9(28) +1 = 1037

1037 is the sum of two squares two different ways:
29² + 14² = 1037
26² + 19² = 1037

1037 is the hypotenuse of FOUR Pythagorean triples:
187-1020-1037 which is 17 times (11-60-61)
315-988-1037 calculated from 26² – 19², 2(26)(19), 26² + 19²
488-915-1037 which is (8-15-17) times 61
645-812-1037 calculated from 29² – 14², 2(29)(14), 29² + 14²

1036 Look, Look to the Rainbow

Finian’s Rainbow is a wonderful movie to enjoy on Saint Patrick’s Day. One of its songs reminds us to “Look, look to the rainbow”.

If you look to this rainbow, you will find all the factors of 1036:

There is a simple symmetry in every rainbow. There is also symmetry in palindromes which are numbers, words, or sentences that read the same forward or backward.

1036 demonstrates that symmetry when it is written in some other bases:
It’s repdigit 4444 in BASE 6 because 4(6³ + 6² + 6¹ + 6⁰) = 4(216 + 36 + 6 + 1) = 4(259) = 1036,
It’s 232 in BASE 22 because 2(22²) + 3(22) + 2(1) = 1036,
1M1 in BASE 23 (M is 22 base 10) because 23² + 22(23) + 1 = 1036, and
SS in BASE 36 (S is 28 base 10) because 28(36 + 1) = 28(37) = 1036

1036 is also the hypotenuse of a Pythagorean triple:
336-980-1036 which is 28 times 12-35-37

  • 1036 is a composite number.
  • Prime factorization: 1036 = 2 × 2 × 7 × 37, which can be written 1036 = 2² × 7 × 37
  • 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 1036 has exactly 12 factors.
  • Factors of 1036: 1, 2, 4, 7, 14, 28, 37, 74, 148, 259, 518, 1036
  • Factor pairs: 1036 = 1 × 1036, 2 × 518, 4 × 259, 7 × 148, 14 × 74, or 28 × 37,
  • Taking the factor pair with the largest square number factor, we get √1036 = (√4)(√259) = 2√259 ≈ 32.18695

There wasn’t a pot of gold at the end of our factor rainbow, but there is one here at the end of this post. It’s a level 5 puzzle, but it isn’t too difficult, so see if you can find all the factors that make the puzzle function like a multiplication table.

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

 

1035 is the 23rd Hexagonal Number

1035 is the 23rd hexagonal number because of the way that it can be factored:
2(23²) – 23 = 1035,
(2(23) – 1)23 = 1035
or simply 45(23) = 1035.

Starting at the bottom of the hexagonal we see 1 yellow square, 5 green squares, 9 blue squares, 13 purple squares, 17 red squares, and 21 orange squares.

1, 5, 9, 13, 17, 21, . . . is an arithmetic progression or arithmetic sequence. The common difference between the numbers is 4.

The nth hexagonal number is the sum of the first n numbers in that arithmetic progression.
The first few hexagonal numbers form an arithmetic series: 1, 6, 15, 28, 45, 66 and so forth.
1035 is the sum of the first 23 numbers in the progression so it is the 23rd term in the series and the 23rd hexagonal number.

All hexagonal numbers are also triangular numbers. 1035 is the 45th triangular number because 45(46)/2 = 1035.

Starting in the lower left-hand corner of that triangle we see 1 yellow square, 2 green squares, 3 blue squares, 4 purple squares, 5 red squares, and 6 orange squares.

1, 2, 3, 4, 5, 6, . . .  is the simplest arithmetic progression there is. The common difference is 1.

The nth triangular number is the sum of the first n numbers in that arithmetic progression.

The first few triangular numbers form an arithmetic series: 1, 3,  6, 10, 15, 21, 28, 36, 45, 55, 66 and so forth.  (The blue triangular numbers are also hexagonal numbers.)
1035 is the sum of the first 45 numbers in the progression so it is the 45th term in the series and the 45th triangular number.

1035 is also the hypotenuse of one Pythagorean triple:
621-828-1035 which is (3-4-5) times 207

It is also a leg in several Pythagorean triples including
1035-1380-1725 which is (3-4-5) times 345

  • 1035 is a composite number.
  • Prime factorization: 1035 = 3 × 3 × 5 × 23, which can be written 1035 = 3² × 5 × 23
  • 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 = 23. Therefore 1035 has exactly 12 factors.
  • Factors of 1035: 1, 3, 5, 9, 15, 23, 45, 69, 115, 207, 345, 1035
  • Factor pairs: 1035 = 1 × 1035, 3 × 345, 5 × 207, 9 × 115, 15 × 69, or 23 × 45,
  • Taking the factor pair with the largest square number factor, we get √1035 = (√9)(√115) = 3√115 ≈ 32.1714