Puzzle

1053 Find the Factors Challenge

/ ivasallay / Leave a comment

If you can multiply and divide and THINK, then you can solve this puzzle. Go ahead. Give it a try! To solve it, write the numbers from 1 to 10 in each of the four bold areas on the puzzle so that the given clues will be the products of those corresponding factors. Good luck!

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

What can I tell you about the number 1053?

  • 1053 is a composite number.
  • Prime factorization: 1053 = 3 × 3 × 3 × 3 × 13, which can be written 1053 = 3⁴ × 13
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1053 has exactly 10 factors.
  • Factors of 1053: 1, 3, 9, 13, 27, 39, 81, 117, 351, 1053
  • Factor pairs: 1053 = 1 × 1053, 3 × 351, 9 × 117, 13 × 81, or 27 × 39
  • Taking the factor pair with the largest square number factor, we get √1053 = (√81)(√13) = 9√13 ≈ 32.44996

27² + 18²  = 1053 so 1053 is the hypotenuse of a Pythagorean triple:
405-972-1053 calculated from 27² – 18², 2(27)(18), 27² + 18²

1053 is the sum of three consecutive powers of 3:
3⁴ + 3⁵ + 3⁶ = 1053

1053 is 3033 in BASE 7 because 3(7³ + 7¹ + 7º) = 3(351) = 1053, and
it’s palindrome 878 in BASE 11 because 8(121) + 7(11) + 8(1) = 1053

Advertisements

1052 A Mysterious Purple Cat for Josephine

/ ivasallay / Leave a comment

My very brilliant friend, Josephine likes cats and her favorite color is purple. Hence, this is a purple cat puzzle made especially for her. It’s a mystery level puzzle so its difficulty level is a big secret. This puzzle only requires skills in multiplication and division. I’m sure it will be no match for Josephine, who can easily handle more advanced mathematics such as calculus. Josephine is also fluent in English, Chinese, Spanish, French, Arabic, and Tajiki.  She is very busy, so hopefully, she’ll be able to find the time to spend some time with this mysterious purple cat.

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

If the colors in the puzzle distract you, here is the same puzzle in very plain black and white:

Here is some information about the number 1052:

Its last two digits are 52, so it is divisible by 4.

  • 1052 is a composite number.
  • Prime factorization: 1052 = 2 × 2 × 263, which can be written 1052 = 2² × 263
  • 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 1052 has exactly 6 factors.
  • Factors of 1052: 1, 2, 4, 263, 526, 1052
  • Factor pairs: 1052 = 1 × 1052, 2 × 526, or 4 × 263
  • Taking the factor pair with the largest square number factor, we get √1052 = (√4)(√263) = 2√263 ≈ 32.434549480453709548485

1052 is palindrome 282 in BASE 21 because 2(21²) + 8(21) + 2(1) = 1052

1049 and Level 6

/ ivasallay / Leave a comment

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.

 

 

1048 and Level 5

/ ivasallay / Leave a comment

What is the common factor needed for 8 and 24 to make this puzzle work? Is it 2, 4, or 8? How about for 12 and 36? Is it 3, 4, 6, or 12? Don’t guess which common factor to use. In each case, all but one of the choices will be eliminated by using logic. It won’t be easy, but if you are determined, you can solve this puzzle.

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

Here are some facts about the number 1048:

  • 1048 is a composite number.
  • Prime factorization: 1048 = 2 × 2 × 2 × 131, which can be written 1048 = 2³ × 131
  • The exponents in the prime factorization are 1 and 3. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1048 has exactly 8 factors.
  • Factors of 1048: 1, 2, 4, 8, 131, 262, 524, 1048
  • Factor pairs: 1048 = 1 × 1048, 2 × 524, 4 × 262, or 8 × 131
  • Taking the factor pair with the largest square number factor, we get √1048 = (√4)(√262) = 2√262 ≈ 32.372828

1048 can be written as the difference of two squares two different ways:
263² – 261² = 1048
133² – 129² = 1048

1048 can also be expressed as 2 times a factor pair 3 different ways:
2(524)(1)
2(262)(2)
2(131)(4)

Those facts make 1048 a leg in these FIVE obscure Pythagorean triples:
1048-137286-137290 calculated from 263² – 261², 2(263)(261), 263² + 261²
1048-34314-34330 calculated from 133² – 129², 2(133)(129), 133² + 129²
1048-274575-274577 calculated from 2(524)(1), 524² – 1², 524² + 1²
1048-68640-68648 calculated from 2(262)(2), 262² – 2², 262² + 2²
1048-17145-17177 calculated from 2(131)(4), 131² – 4², 131² + 4²

1047 and Level 4

/ ivasallay / Leave a comment

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

/ ivasallay / Leave a comment

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

1045 and Level 2

/ ivasallay / Leave a comment

This puzzle consists of six sets of three numbers. Find the common factor of each set of clues so that ALL of the factors involved are a number from 1 to 12, and you’ll solve this puzzle. Have fun!

Here are a few facts about the number 1045:

Obviously, 1045 can be evenly divided by 5, but since 1-0+4-5 = 0, we know that 1045 is also divisible by 11.

  • 1045 is a composite number.
  • Prime factorization: 1045 = 5 × 11 × 19
  • 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 1045 has exactly 8 factors.
  • Factors of 1045: 1, 5, 11, 19, 55, 95, 209, 1045
  • Factor pairs: 1045 = 1 × 1045, 5 × 209, 11 × 95, or 19 × 55
  • 1045 has no square factors that allow its square root to be simplified. √1045 ≈ 32.32646

1045 is the hypotenuse of a Pythagorean triple:
627-836-1045 which is (3-4-5) times 209.

1045 is also palindrome 171 in BASE 29 because 1(29²) + 7(29) + 1(1) = 1045

1044 and Level 1

/ ivasallay / Leave a comment

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

1043 Find the Factors Challenge Puzzle

/ ivasallay / Leave a comment

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

/ ivasallay / Leave a comment

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