How Can You Count These 1054 Tiny Squares?

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There are 1054 tiny squares in the image below, making 1054 a centered triangular number. How can you know that I’m not pulling the wool over your eyes about the number of tiny squares? Here are a few ways that you can quickly count all of them.

If you start with the yellow square in the center and count outward each succeeding triangle you will get 1 yellow square + 3 green squares + 6 blue squares + 9 purple squares + 12 red squares + 15 orange squares, etc. until you reach the final 78 blue squares:
1 + 3 + 6 + 9 + 12 + 15 + . . . + 78
= 1 + 3(1 + 2 + 3 + 4 + 5 + . . . + 26)
= 1 + 3(26*27)/2 = 1 + 3(351) = 1054

Using a little bit of algebra, you can show that
1 + 3(26*27)/2 = (3(26²) + 3(26) + 2)/2 = 1054

You can divide the centered triangle above into three triangles as I also did in the graphic. The three triangles represent the 25th, the 26th, and the 27th triangular numbers. Adding them up you get:
25(26)/2 + 26(27)/2 + 27(28)/2 = 325 + 351 + 378 = 1054

Here is some more information about the number 1054:

  • 1054 is a composite number.
  • Prime factorization: 1054 = 2 × 17 × 31
  • 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 1054 has exactly 8 factors.
  • Factors of 1054: 1, 2, 17, 31, 34, 62, 527, 1054
  • Factor pairs: 1054 = 1 × 1054, 2 × 527, 17 × 62, or 31 × 34
  • 1054 has no square factors that allow its square root to be simplified. √1054 ≈ 32.465366

1054 is the sum of six consecutive prime numbers:
163 + 167 + 173 + 179 + 181 + 191 = 1054

1054 is the hypotenuse of a Pythagorean triple:
496-930-1054 which is (8-15-17) times 62

1054 looks interesting when it is written in some other bases:
It’s 4A4 in BASE 15 (A is 10 base 10) because 4(15²) + 10(15) + 4(1) = 1054
1C1 in BASE 27 (C is 12 base 10) because 27² + 12(27) + 1 = 1054
VV in BASE 33 (V is 31 base 10) because 31(33) + 31(1) = 31(34) = 1054
V0 in BASE 34 because 31(34) = 1054

 

 

 

 

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1053 Find the Factors Challenge

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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

1052 A Mysterious Purple Cat for Josephine

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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

1051 is the 21st Centered Pentagonal Number

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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

1050 Factor Trees Are In Bloom This Spring

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1050 has much more than the average number of factors for a number its size. I decided to make a few of its MANY possible factor trees in some springtime colors. I hope you enjoy looking at them. Its prime factors, 2, 3, 5, 5, and 7, are in green.

What can I tell you about the number 1050?

  • 1050 is a composite number.
  • Prime factorization: 1050 = 2 × 3 × 5 × 5 × 7, which can be written 1050 = 2 × 3 × 5² × 7
  • The exponents in the prime factorization are 1, 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(2 + 1)(1 + 1) = 2 × 2 × 3 × 2 = 24. Therefore 1050 has exactly 24 factors.
  • Factors of 1050: 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 25, 30, 35, 42, 50, 70, 75, 105, 150, 175, 210, 350, 525, 1050
  • Factor pairs: 1050 = 1 × 1050, 2 × 525, 3 × 350, 5 × 210, 6 × 175, 7 × 150, 10 × 105, 14 × 75, 15 × 70, 21 × 50, 25 × 42, or 30 × 35
  • Taking the factor pair with the largest square number factor, we get √1050 = (√25)(√42) = 5√42 ≈ 32.403703

If you had $10.50 in quarters, you would have 42 quarters because 42(25) = 1050
If you had $10.50 in dimes, you would have 105 dimes because 105(10) = 1050
If you had $10.50 in nickles, you would have 210 nickles because 210(5) = 1050

1050 is the sum of the twenty-two primes from 7 to 97.

1050 is the hypotenuse of two Pythagorean triples:
630-840-1050 which is (3-4-5) times 210
294-1008-1050 which is (7-24-25) times 42

1050 looks interesting to me when it is written in a couple of other bases:
It’s UU in BASE 34 (U is 30 base 10) because 30(34) + 30(1) = 30(35) = 1050,
and it’s U0 in BASE 35 because 30(35) = 1050

1049 and Level 6

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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

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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

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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

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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

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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