1033 and Level 3

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To solve a level 3 puzzle begin with 80, the clue at the very top of the puzzle. Clue 48 goes with it. What are the factor pairs of those numbers in which both factors are between 1 and 12 inclusive? 80 can be 8×10, and 48 can be 4×12 or 6×8. What is the only number that listed for both 80 and 48? Put that number in the top row over the 80. Put the corresponding factors where they go starting at the top of the first column.

Work down that first column cell by cell finding factors and writing them as you go. Three of the factors have been highlighted because you have to at least look at the 55 and the 5 to deal with the 20 in the puzzle. Have fun!

Print the puzzles or type the solution in this excel file: 12 factors 1028-1034

Here are a few facts about the number 1033:

It is a twin prime with 1031.

32² + 3² = 1033, so it is the hypotenuse of a Pythagorean triple:
192-1015-1033 calculated from 2(32)(3), 32² – 3², 32² + 3²

1033 is a palindrome in two other bases:
It’s 616 in BASE 13 because 6(13²) + 1(13) + 6(1) = 1033
1J1 in BASE 24 (J is 19 base 10) because 24² + 19(24) + 1 = 1033

8¹ + 8º + 8³ + 8³ = 1033 Thanks to OEIS.org for that fun fact!

  • 1033 is a prime number.
  • Prime factorization: 1033 is prime.
  • The exponent of prime number 1033 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1033 has exactly 2 factors.
  • Factors of 1033: 1, 1033
  • Factor pairs: 1033 = 1 × 1033
  • 1033 has no square factors that allow its square root to be simplified. √1033 ≈ 32.1403

How do we know that 1033 is a prime number? If 1033 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1033 ≈ 32.1. Since 1033 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1033 is a prime number.

Here’s another way we know that 1033 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 32² + 3² = 1033 with 32 and 3 having no common prime factors, 1033 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1033 ≈ 32.1. Since 1033 is not divisible by 5, 13, 17, or 29, we know that 1033 is a prime number.

 

1032 How Many Twelves Are in a 12×12 Times Table?

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How many 12’s are in a standard 12×12 times table? There are five 12’s in the puzzle below. Six, if you count the 12 in the title. Is that too many, just the right number, or are there even more?

This is only a level 2 puzzle so it won’t be difficult to solve. . . unless I’ve put in too many 12’s!

Hmm. . .Try solving the puzzle, then fill in the rest of the multiplication table. Then you will know for sure how many 12’s SHOULD be in the table.

Print the puzzles or type the solution in this excel file: 12 factors 1028-1034

The number 1032 is divisible by 12. Here are a few more facts about that number:

1032 is made with a zero and three consecutive numbers so it is divisible by 3.

The last two digits of 1032 are 32 so 1032 can be evenly divided by 4.

Since 32 is divisible by 8 and preceded by an even zero in 1032, our number is also divisible by 8.

As you will soon see, 1032 is divisible by even more numbers than those listed above. Here are three of its factor trees:

1032 looks interesting in a couple of different bases:
It’s 4440 in BASE 6 because 4(6³ + 6² + 6¹) = 4(258) = 1032, and
it’s palindrome 3003 in BASE 7 because 3(7³ + 7⁰) = 3(344) = 1032.

  • 1032 is a composite number.
  • Prime factorization: 1032 = 2 × 2 × 2 × 3 × 43, which can be written 1032 = 2³ × 3 × 43
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 1032 has exactly 16 factors.
  • Factors of 1032: 1, 2, 3, 4, 6, 8, 12, 24, 43, 86, 129, 172, 258, 344, 516, 1032
  • Factor pairs: 1032 = 1 × 1032, 2 × 516, 3 × 344, 4 × 258, 6 × 172, 8 × 129, 12 × 86, or 24 × 43
  • Taking the factor pair with the largest square number factor, we get √1032 = (√4)(√258) = 2√258 ≈ 32.12476

1031 Prepare for World Maths Day 2018

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You might think that a day lasts 24 hours, but strategic use of the international date line can actually make a single day last 48 hours!

How will you spend the 48 hour day that will be 7 March 2018?

Colleen Young encourages you and your class to register for and participate in World Maths Day 2018 held that day. She shares the necessary links as well as several tips on how to prepare.

One way to prepare now is playing multiplication games like the level 1 puzzle below. The puzzle is just a multiplication table but the factors are missing and only a few of the products are given, and they aren’t in the order you would normally expect. Can you figure out where the factors from 1 to 12 belong in both the first column and the top row of the puzzle?

Print the puzzles or type the solution in this excel file: 12 factors 1028-1034

If this puzzle is too easy for you, Then it is time to move on to a level 2 or higher puzzle. You can find one in the link above and plenty others here at findthefactors.com.

Now I’d like to tell you some things that I’ve learned about the number 1031:

1031 and 1033 are twin primes.

1031 is a palindrome in a couple of bases:
It’s 858 in BASE 11 because 8(121) + 5(11) + 8(1) = 1031 and
it’s 272 in BASE 21 because 2(441) + 7(21) + 2(1) = 1031

  • 1031 is a prime number.
  • Prime factorization: 1031 is prime.
  • The exponent of prime number 1031 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1031 has exactly 2 factors.
  • Factors of 1031: 1, 1031
  • Factor pairs: 1031 = 1 × 1031
  • 1031 has no square factors that allow its square root to be simplified. √1031 ≈ 32.10919

How do we know that 1031 is a prime number? If 1031 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1031 ≈ 32.1. Since 1031 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1031 is a prime number.

1030 Cupid’s Arrow

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Love can often be like a puzzle. When cupid’s arrow hits its mark, at first everything might seem to fall into place, but before long, love starts getting complicated and has to be figured out.

That’s the way this Cupid’s Arrow puzzle is, too. It’s easy to find the logic to start it, but then the logic will be more difficult to see. May you be able to figure out this puzzle as well as the important relationships in your life!

Print the puzzles or type the solution in this excel file: 12 factors 1028-1034

What can I tell you about the number 1030?

It’s the sum of two consecutive prime numbers:
509 + 521 = 1030

It’s the hypotenuse of a Pythagorean triple:
618-824-1030 which is (3-4-5) times 206

It’s palindrome 1102011 in BASE 3 because 3⁶ + 3⁵ + 2(3³) + 3¹ + 3⁰ = 1030

  • 1030 is a composite number.
  • Prime factorization: 1030 = 2 × 5 × 103
  • 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 1030 has exactly 8 factors.
  • Factors of 1030: 1, 2, 5, 10, 103, 206, 515, 1030
  • Factor pairs: 1030 = 1 × 1030, 2 × 515, 5 × 206, or 10 × 103
  • 1030 has no square factors that allow its square root to be simplified. √1030 ≈ 32.09361

 

1029 A Rose for Your Valentine

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A dozen roses can be pretty pricey around Valentine’s Day, but at least one website asserts that a single rose can make just as big a statement and just as big an impact. Today’s mystery level puzzle looks like a single rose. I hope you will enjoy its beauty even if its thorns are prickly.

Print the puzzles or type the solution in this excel file: 12 factors 1028-1034

Here are a few facts about the number 1029:

It’s easy to see that 1029 can be evenly divided by 3 because 1 + 0 + 2 + 9 = 12, a number divisible by 3.

It’s not quite as easy to tell that it can be evenly divided by 7:
It is because 102 – 2(9) = 102 – 18 = 84, a number divisible by 7.

I like the way 1029 looks when it is written in some other bases:
It’s 4433 in BASE 6 because 4(6³) + 4(6²) + 3(6¹) + 3(6⁰) = 4(216 + 36) + 3(6 + 1) = 1029,
3000 in BASE 7 because 3(7³) = 3(343) = 1029,
399 in BASE 17 because 3(17²) + 9(17) + 9(1) = 3(289 + 51 + 3) = 3(343) = 1029
333 in BASE 18 because 3(18² + 18 + 1) = 3(343) = 1029

  • 1029 is a composite number.
  • Prime factorization: 1029 = 3 × 7 × 7 × 7, which can be written 1029 = 3 × 7³
  • The exponents in the prime factorization are 1 and 3. Adding one to each and multiplying we get (1 + 1)(3 + 1) = 2 × 4 = 8. Therefore 1029 has exactly 8 factors.
  • Factors of 1029: 1, 3, 7, 21, 49, 147, 343, 1029
  • Factor pairs: 1029 = 1 × 1029, 3 × 343, 7 × 147, or 21 × 49
  • Taking the factor pair with the largest square number factor, we get √1029 = (√49)(√21) = 7√21 ≈ 32.07803

 

1028 A Valentine Mystery

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Valentine’s Day is almost here so I’ve made three Valentine related puzzles this week. I’ve labeled all of three of them Mystery Level because you might find some of them to be difficult. Use logic and an ordinary 12 × 12 multiplication table. I promise that each one of them can be solved, and I hope that you LOVE working on them! There will be some easier puzzles later on in the week.

Print the puzzles or type the solution in this excel file: 12 factors 1028-1034

Let me tell you a little about the number 1028:

32² + 2² = 1028 so 1028 is the hypotenuse of a Pythagorean triple:
128-1020-1028 which is 4 times (32-255-257) and can be calculated from 2(32)(2), 32² – 2², 32² + 2²

1028 is a palindrome when it is written in a couple of different bases:
404 in BASE 16 because 4(16²) + 4(1) = 4(257) = 1028
2G2 in BASE 19 (G is 16 base 10) because 2(19²) + 16(19) + 2(1) = 1028

  • 1028 is a composite number.
  • Prime factorization: 1028 = 2 × 2 × 257, which can be written 1028 = 2² × 257
  • 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 1028 has exactly 6 factors.
  • Factors of 1028: 1, 2, 4, 257, 514, 1028
  • Factor pairs: 1028 = 1 × 1028, 2 × 514, or 4 × 257
  • Taking the factor pair with the largest square number factor, we get √1028 = (√4)(√257) = 2√257 ≈ 32.062439

 

1027 Find the Factors Challenge

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I’m really enjoying these Find the Factors Challenge puzzles, and I hope that you will give them a try and love them, too. You can find this one as well as a little less challenging one in the excel file link below the puzzle. You can type the factors directly on that file. Remember to use logic for every single factor pair you use.

Print the puzzles or type the solution in this excel file: 10-factors-1019-1027

Now I’ll tell you a little about the number 1027:

It is the sum of seven consecutive prime numbers:
131 + 137 + 139 + 149 + 151 + 157 + 163 = 1027

It is the sum of the squares of the first eight prime numbers:
2² +  3² +  5² +  7² +  11² +  13² +  17² +  19² = 1027
Indeed, 666 + 19² = 1027. Thanks to OEIS.org for that fun fact.

Because 19³ – 18³ = 1027, it is the 19th Centered Hexagonal Number.
That also means that 19² + 19(18) + 18² = 1027
because a³ – b³ = (a-b)(a²+ab+b²).

1027 is the hypotenuse of a Pythagorean triple:
395-948-1027 which is (5-12-13) times 79.

1027 is also a palindrome when it is written in these three other bases:
717 in BASE 12
535 in BASE 14
1B1 in BASE 27 (B is 11 base 10)

  • 1027 is a composite number.
  • Prime factorization: 1027 = 13 × 79
  • 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 1027 has exactly 4 factors.
  • Factors of 1027: 1, 13, 79, 1027
  • Factor pairs: 1027 = 1 × 1027 or 13 × 79
  • 1027 has no square factors that allow its square root to be simplified. √1027 ≈ 32.0468

 

1026 One Last Mystery

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I made a week’s worth of mystery level puzzles, and today’s puzzle is the last one in the set. Find the Factors of the clues in this puzzle by using logic and your knowledge of the multiplication table. It may not be easy for you, but give it a try anyway. If you find it too difficult, I will soon publish some more easier-level puzzles.

Print the puzzles or type the solution in this excel file: 10-factors-1019-1027

Now let me share a few facts about the number 1026:

1026 is the sum of the fourteen prime numbers from 43 to 103.

I like the way 1026 looks when it is written in these other bases:
It’s 2002 in BASE 8,
396 in BASE 17,
330 in BASE 18,
1G1 in BASE 25 (G is 16 base 10), and
123 in BASE 31

  • 1026 is a composite number.
  • Prime factorization: 1026 = 2 × 3 × 3 × 3 × 19, which can be written 1026 = 2 × 3³ × 19
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 × 4 × 2 = 16. Therefore 1026 has exactly 16 factors.
  • Factors of 1026: 1, 2, 3, 6, 9, 18, 19, 27, 38, 54, 57, 114, 171, 342, 513, 1026
  • Factor pairs: 1026 = 1 × 1026, 2 × 513, 3 × 342, 6 × 171, 9 × 114, 18 × 57, 19 × 54, or 27 × 38
  • Taking the factor pair with the largest square number factor, we get √1026 = (√9)(√114) = 3√114 ≈ 32.03123

1025 Mystery Date

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It is a mystery why we in the United States write our dates “month-day-year”. It makes about as much sense as saying larger, large, largest or better, good, best. Nevertheless, it is what it is.

So today in the United States it is 2-7-18, the e-day of the century. It’s not quite as exciting as 2-7-1828 might have been, but still pretty exciting. e is also known as Euler’s number, and like pi, it is an irrational number. A college professor of mine taught me how to remember its first few digits by remembering 2.7, the year 1828 twice, and 45-90-45 (that very important isosceles triangle). Thus, e ≈ 2.718281828459045.

The difficulty level of today’s puzzle is also a mystery. Nevertheless, you can still solve it by applying logic and facts from a simple 10×10 multiplication table:

Print the puzzles or type the solution in this excel file: 10-factors-1019-1027

Here are a few facts about the number 1025:

1025 can be written as the sum of consecutive prime numbers two different ways:
97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 = 2025; that’s nine consecutive prime numbers.
71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 = 2025; that’s eleven consecutive prime numbers.

1025 is the sum of two squares three different ways:
25² + 20² = 1025
32² + 1² = 1025
31² + 8² = 1025

That previous fact contributes to the fact that 1025 is the hypotenuse of SEVEN Pythagorean triples:
64-1023-1025 calculated from 2(32)(1), 32² – 1², 32² + 1²
225-1000-1025 which is 25 times (9-40-41) and can also be calculated from 25² – 20², 2(25)(20), 25² + 20²
287-984-1025 which is (7-24-25) times 41
420-935-1025 which is 5 times (84-187-205)
496-897-1025 calculated from 2(31)(8), 31² + 8², 31² + 8²
615-820-1025 which is (3-4-5) times 205
665-780-1025 which is 5 times (133-156-205)

1025 is also a wonderful palindrome in three different bases.
10000000001 in BASE 2
100001 in BASE 4
101 in BASE 32

  • 1025 is a composite number.
  • Prime factorization: 1025 = 5 × 5 × 41, which can be written 1025 = 5² × 41
  • 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 1025 has exactly 6 factors.
  • Factors of 1025: 1, 5, 25, 41, 205, 1025
  • Factor pairs: 1025 = 1 × 1025, 5 × 205, or 25 × 41
  • Taking the factor pair with the largest square number factor, we get √1025 = (√25)(√41) = 5√41 ≈ 32.01562

 

What’s Special About √1024?

/ ivasallay / 2 Comments

What’s special about √1024? Is it because it and several counting numbers after it have square roots that can be simplified?

Perhaps.

Maybe it is interesting just because √1024 = 32, a whole number. The 5th root of 1024 = 4 and the 10th root of 1024 = 2, both whole numbers as well.

Those equations are true because 32² = 1024, 4⁵ = 1024, and 2¹⁰ = 1024.

Or perhaps 1024 is special because it is the smallest number that is a 10th power. (It is 2¹⁰.) The square root of a perfect 10th power is always a perfect 5th power. (32 = 2⁵ and is the smallest number that is a 5th power.)

1024 is also the smallest number with exactly 11 factors.

It is the smallest number whose factor tree has at least 10 leaves that are prime numbers. (They are the red leaves on the factor tree shown below.) It is possible to draw several other factor trees for 1024, but they will all have the number 2 appearing ten times.

What’s more, I noticed something about 1024 and some other multiples of 256: Where do multiples of 256 fall on the list of square roots that can be simplified?

  • 256 × 1 = 256 and 256 is the 100th number on this list of numbers whose square roots can be simplified.

1st 100 reducible square roots

  • 256 × 2 = 512. When we add the next 100 square roots that can be simplified, 512 is the 199th number on the list.

2nd 100 reducible square roots

  • Here are the third 100 square roots that can be simplified:

Reducible Square Roots 516-765

  • 256 × 3 = 768 didn’t quite make that list because it is the 301st number. Indeed, it is the first number on this list of the fourth 100 numbers whose square roots can be simplified.

  • 256 × 4 = 1024. That will be the first number on the 5th 100 square roots list!

It is interesting that those multiples of 256 have the 100th, the 199th, the 301st, and the 401st positions on the list. That is so close to the 100th, 200th, 300th, and 400th positions.

In case you couldn’t figure it out, the highlighted square roots are three or more consecutive numbers that appear on the list.

1024 is interesting for many other reasons. Here are a few of them:

(4-2)¹⁰ = 1024, making 1024 the 16th Friedman number.

I like to remember that 2¹⁰ = 1024, which is just a little bit more than a thousand. Likewise 2²⁰ = 1,048,576 which is about a million. 2³⁰ is about a billion, and 2⁴⁰ is about a trillion.

*******
As stated in the comments, Paula Beardell Krieg shared a related post with me. It takes exactly 1024 Legos to build this fabulous pyramidal fractal:

https://platform.twitter.com/widgets.js
1024 has so many factors that are divisible by 4 that it is a leg in NINE Pythagorean triples:
768-1024-1280 which is (3-4-5) times 256
1024-1920-2176 which is (8-15-17) times 128
1024-4032-4160 which is (16-63-65) times 64
1024-8160-8224 which is (32-255-257) times 32
1024-16368-16400 which is 16 times (64-1023-1025)
1024-32760-32776 which is 8 times (128-4095-4097)
1024-65532-65540 which is 4 times (256-16383-16385)
1024-131070-131074 which is 2 times (512-65535-65537),
and primitive 1024-262143-262145

Some of those triples can also be found because 1024 is the difference of two squares four different ways:
257² – 255² = 1024
130² – 126² = 1024
68² – 60² = 1024
40² – 24² = 1024
To find out which difference of two squares go with which triples, add the squares instead of subtracting and you’ll get the hypotenuse of the triple.
******

1024 looks interesting in some other bases:
It’s 1000000000 in BASE 2,
100000 in BASE 4,
2000 in BASE 8,
1357 in BASE 9,
484 in BASE 15,
400 in BASE 16,
169 in BASE 29,
144 in BASE 30,
121 in BASE 31, and
100 in BASE 32

  • 1024 is a composite number.
  • Prime factorization: 1024 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2, which can be written 1024 = 2¹⁰
  • The exponent in the prime factorization is 10. Adding one we get (10 + 1) = 11. Therefore 1024 has exactly 11 factors.
  • Factors of 1024: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
  • Factor pairs: 1024 = 1 × 1024, 2 × 512, 4 × 256, 8 × 128, 16 × 64, or 32 × 32,
  • 1024 is a perfect square. √1024 = 32. It is also a perfect 5th power, and a perfect 10th power.