Author: ivasallay

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.

Mysterious 1023

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2⁹ + 2⁸ + 2⁷ + 2⁶ + 2⁵ + 2⁴ + 2³ + 2² + 2¹ + 2⁰ = 1023. That makes 1023 a pretty cool and rather mysterious number.

This puzzle that I’ve numbered 1023 is pretty cool and mysterious, too. I’m sure you will enjoy solving it if you only use logic to find the solution.

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

Here are some other fascinating facts about the number 1023:

It is formed by using a zero and three other consecutive numbers, so it is divisible by 3.

1 – 0 + 2 – 3 = 0, so 1023 is divisible by eleven.

31 × 33 = 1023 so (32 – 1)(32 + 1) = 1023, AND it is 32² – 1, making it one away from the next square number!

It is the sum of five consecutive prime numbers:
193 + 197 + 199 + 211 + 223 = 1023

1023 looks quite interesting when it is written in several different bases:
First of all, it’s 1111111111 in BASE 2 because it is the sum of the all those powers of 2 from 0 to 9 that were included at the top of this post.

It’s also 33333 in BASE 4 because 3(4⁴ + 4³ + 4² + 4¹ + 4⁰) = 3(341) = 1023.
That also means that 3(2⁸ + 2⁶ + 2⁴ + 2² + 2⁰) = 1023

It’s 393 in BASE 17 because 3(17²) + 9(17) + 3(1) = 1023,
VV in BASE 32 (V is 31 base 10) because 31(32) + 31(1) = 31(33) = 1023, and
V0 in BASE 33 because 31(33) = 1023

  • 1023 is a composite number.
  • Prime factorization: 1023 = 3 × 11 × 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 1023 has exactly 8 factors.
  • Factors of 1023: 1, 3, 11, 31, 33, 93, 341, 1023
  • Factor pairs: 1023 = 1 × 1023, 3 × 341, 11 × 93, or 31 × 33
  • 1023 has no square factors that allow its square root to be simplified. √1023 ≈ 31.98437

1022 Friedman Number Mystery

/ ivasallay / 2 Comments

1022 is the 15th Friedman number. “What is a Friedman number and why is 1022 one of them?” you may ask. I will solve that little mystery for you. 1022 is a Friedman number because
2¹⁰ – 2 = 1022. Notice that the expression 2¹⁰ – 2 uses the digits 1, 0, 2, and 2 in some order and a subtraction sign. A Friedman number can be written as an expression that uses all of its own digits the exact number of times that they occur in the number. The expression must include at least one operator (+, -, ×, ÷) or a power. Parenthesis are allowed as long as the other rules are followed.

Now I would like you to solve the mystery of this puzzle using logic and the multiplication facts. Can you do it?

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

1022 is the hypotenuse of a Pythagorean triple:
672-770-1022 which is 14 times (48-55-73)

  • 1022 is a composite number.
  • Prime factorization: 1022 = 2 × 7 × 73
  • 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 1022 has exactly 8 factors.
  • Factors of 1022: 1, 2, 7, 14, 73, 146, 511, 1022
  • Factor pairs: 1022 = 1 × 1022, 2 × 511, 7 × 146, or 14 × 73
  • 1022 has no square factors that allow its square root to be simplified. √1022 ≈ 31.96873

1021 Mystery Level

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Here is the second puzzle in a week’s worth of mystery level puzzles. Will it be very difficult or not so bad? That’s the mystery. You’ll have to try it to know for sure. You only need to use logic and your knowledge of the multiplication table to solve it. I wish you luck!

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

That puzzle may have been a mystery, but the number 1021 isn’t much of a mystery at all. It is the second number of twin primes, 1019 and 1021.

Since it is a prime number, and it has a remainder of one when it is divided by 4, it can be written as the sum of two squares:
30² + 11² = 1021.

Since it can be written as the sum of two squares, it is the hypotenuse of a Pythagorean triple:
660-779-1021 calculated from 2(30)(11), 30² – 11², 30² + 11²

It is also palindrome 141 in BASE 30 because 1(30²) + 4(30) + 1(1) = 1021

  • 1021 is a prime number.
  • Prime factorization: 1021 is prime.
  • The exponent of prime number 1021 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1021 has exactly 2 factors.
  • Factors of 1021: 1, 1021
  • Factor pairs: 1021 = 1 × 1021
  • 1021 has no square factors that allow its square root to be simplified. √1021 ≈ 31.95309

How do we know that 1021 is a prime number? If 1021 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1021 ≈ 31.95. Since 1021 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1021 is a prime number.

Here’s another way we know that 1021 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 30² + 11² = 1021 with 30 and 11 having no common prime factors, 1021 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1021 ≈ 31.95. Since 1021 is not divisible by 5, 13, 17, or 29, we know that 1021 is a prime number.

1020 A Week of Mystery

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Sometimes changing things up a little is good. I decided to make a week’s worth of mystery level puzzles. The actual difficulty level will vary from puzzle to puzzle so give each one of them a try. If you think one is too easy or too difficult, the next one might not be. Here’s the first one:

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

Here’s a little bit about the number 1020:

It is only 8 more than 1008, the previous number that also had 24 factors. Here are a few of its MANY possible factor trees.

1020 is the sum of six consecutive prime numbers:
157 + 163 + 167 + 173 + 179 + 181 = 1020

1020 is the hypotenuse of four Pythagorean triples:
156-1008-1020 which is 12 times (13-84-85)
432-924-1020 which is 12 times (36-77-85)
480-900-1020 which is (8-15-17) times 60
612-816-1020 which is (3-4-5) times 204

1020 looks interesting when it is written using some different bases:
It’s 33330 in BASE 4 because 3(4⁴ + 4³ + 4² + 4¹) = 3(340) = 1020,
848 in BASE 11 because 8(11²) + 4(11) + 8(1) = 1020,
606 in BASE 13 because 6(13²) + 6(1) = 6(170) = 1020,
480 in BASE 15 because 4(15²) + 8(15) = 4(225 + 30) = 4(255) = 1020,
390 in BASE 17 because 3(17²) + 9(17) = 3(289 + 51) = 3(340) = 1020,
UU in BASE 33 (U is 30 base 10) because 30(33) + 30(1) = 30(34) = 1020, and
U0 in BASE 34 because 30(34) = 1020

  • 1020 is a composite number.
  • Prime factorization: 1020 = 2 × 2 × 3 × 5 × 17, which can be written 1020 = 2² × 3 × 5 × 17
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1020 has exactly 24 factors.
  • Factors of 1020: 1, 2, 3, 4, 5, 6, 10, 12, 15, 17, 20, 30, 34, 51, 60, 68, 85, 102, 170, 204, 255, 340, 510, 1020
  • Factor pairs: 1020 = 1 × 1020, 2 × 510, 3 × 340, 4 × 255, 5 × 204, 6 × 170, 10 × 102, 12 × 85, 15 × 68, 17 × 60, 20 × 51, or 30 × 34
  • Taking the factor pair with the largest square number factor, we get √1020 = (√4)(√255) = 2√255 ≈ 31.9374

1019 An Easier Find the Factors Challenge Puzzle

/ ivasallay / 4 Comments

I’ve recently posted some more challenging puzzles that I’ve named Find the Factors 1 – 10 Challenge, and they definitely are a more challenging puzzle than one of my more traditional level 6 puzzles. As of today, no one has informed me that they have been able to solve either puzzle number 1000 or 1010.

Two years ago I made perhaps my most challenging level 6 puzzle, a 16 × 16 puzzle to commemorate Steve Morris’s birthday. Steve Morris was the very first person to type a comment on my blog, and I have appreciated his encouragement over the years. Steve has solved many kinds of puzzles in his life including some of the toughest I have made, but the puzzle I made for that birthday was no picnic for even him to complete.

This year I’ve made him a challenging puzzle, but it is still a little easier than the other two challenge puzzles I’ve made. If you’ve tried either of those other puzzles without success, still give this one a try. Good luck to you all, and Happy Birthday to Steve Morris! I saved this post number (1019) for you because it uses your birthdate numbers, howbeit out of order.

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

This is my 1019th post. Here are a few facts about the number 1019.

Prime number 1019 is the sum of the 19 prime numbers from 17 to 97.

  • 1019 is a prime number.
  • Prime factorization: 1019 is prime.
  • The exponent of prime number 1019 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1019 has exactly 2 factors.
  • Factors of 1019: 1, 1019
  • Factor pairs: 1019 = 1 × 1019
  • 1019 has no square factors that allow its square root to be simplified. √1019 ≈ 31.921779

How do we know that 1019 is a prime number? If 1019 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1019 ≈ 31.9. Since 1019 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1019 is a prime number.

1018 and Level 6

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Level 6 puzzles can be tricky to solve, but I promise that you can still solve this one using logic and knowledge of the basic multiplication table. Just write the numbers from 1 to 12 in both the first column and the top row so that the puzzle is like a partially filled out multiplication table with the factors in a different order. Like always, there is only one solution. Can you find it?

Print the puzzles or type the solution in this excel file: 12 factors 1012-1018

Look at these interesting facts about the number 1018:

27² + 17² = 1018
That means that 1018 is the hypotenuse of a Pythagorean triple:
440-918-1018 calculated from 27² – 17², 2(27)(17), 27² + 17²

It also means that (44² – 10²)/2 = 1018
Note that 27 + 17 = 44 and 27 – 17 = 10

1018 is full house 33322 in BASE 4 because 3(4⁴) + 3(4³) + 3(4²) + 2(4¹) + 2(4⁰) = 3(256 + 64 + 16) + 2(4 + 1) = 1018

Since 1018 is the sum of odd squares, it is divisible by 2. Since those odd squares have no common prime factors, you only have to check to see if 1018 is divisible by any Pythagorean triple hypotenuses less than or equal to (√1018)/2 ≈ 15.953. It is not divisible by 5 or 13, therefore 1018 only has two prime factors: 2 and 1018/2.

  • 1018 is a composite number.
  • Prime factorization: 1018 = 2 × 509
  • 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 1018 has exactly 4 factors.
  • Factors of 1018: 1, 2, 509, 1018
  • Factor pairs: 1018 = 1 × 1018 or 2 × 509
  • 1018 has no square factors that allow its square root to be simplified. √1018 ≈ 31.90611

1017 and Level 5

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You might find today’s puzzle to be a little trickier than most level 5 puzzles, but don’t let that deter you from giving it your best effort. For example, it’s true that 6 and 12 are both common factors of 60 and 36, but some of the other clues will eliminate either the 6 or the 12. Can you figure out which one gets eliminated?

Print the puzzles or type the solution in this excel file: 12 factors 1012-1018

Let me share some reasons 1017 is an interesting number.

24² + 21² = 1007

1017 is the hypotenuse of a Pythagorean triple:
135-1008-1017 which is 9 times (15-112-113). It can also be calculated
from 24² – 21², 2(24)(21), 24² + 21²

1017 is also palindrome 1771 in BASE 8 because 1(8³) + 7(8²) + 7(8¹) + 1(8⁰) = 1017

  • 1017 is a composite number.
  • Prime factorization: 1017 = 3 × 3 × 113, which can be written 1017 = 3² × 113
  • 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 1017 has exactly 6 factors.
  • Factors of 1017: 1, 3, 9, 113, 339, 1017
  • Factor pairs: 1017 = 1 × 1017, 3 × 339, or 9 × 113
  • Taking the factor pair with the largest square number factor, we get √1017 = (√9)(√113) = 3√113 ≈ 31.890437

1016 and Level 4

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If you’ve never tried a level 4 puzzle before, this is a great one to try. Of its twelve clues, eight have only one factor pair in which both factors are from 1 to 12. You should easily be able to place the factors for those eight clues. Since each factor from 1 to 12 must appear exactly one time in the first column and the top row, the factors from those eight clues will eliminate some of the possible factors of the other four clues. Don’t be afraid to give this puzzle a try!

Print the puzzles or type the solution in this excel file: 12 factors 1012-1018

Here are some reasons why 1016 is an interesting number:

1016 is the sum of seven consecutive powers of two:
2⁹ + 2⁸ + 2⁷ + 2⁶ + 2⁵ + 2⁴ + 2³ = 1016
I know that’s true because 1016 is 1111111000 in BASE 2

1016 is a palindrome in a couple of bases as well:
It’s 13031 in BASE 5 because 1(5⁴) + 3(5³) + 0(5²) + 3(5¹) + 1(5⁰) = 1016
161 in BASE 29 because 1(29²) + 6(29¹) + 1(29⁰) = 1016

1016 is divisible by 2 because 6 is even.
1016 is divisible by 4 because 16 is divisible by 4. (And also because 6 is divisible by 2 but NOT by 4 and 1 is odd.)
1016 is divisible by 8 because 016 is divisible by 8. (And because 16 is divisible by 8 and 0 is an even number.
1016 is NOT divisible by 16 because 016 is divisible by 16 and 1 is an odd number.

  • 1016 is a composite number.
  • Prime factorization: 1016 = 2 × 2 × 2 × 127, which can be written 1016 = 2³ × 127
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1016 has exactly 8 factors.
  • Factors of 1016: 1, 2, 4, 8, 127, 254, 508, 1016
  • Factor pairs: 1016 = 1 × 1016, 2 × 508, 4 × 254, or 8 × 127
  • Taking the factor pair with the largest square number factor, we get √1016 = (√4)(√254) = 2√254 ≈ 31.87475