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

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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

1015 and Level 3

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If you glance at this puzzle for a few seconds, you may think there are three places in the top row where the number 1 will work, but in actuality, only one of those places will work with all the other clues in the puzzle. This is a level 3 puzzle, so start with the top cell in the first column and work down cell by cell placing factors in both the first column and the top row until the puzzle resembles a multiplication table.

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

Here are a few facts about the number 1015:

1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² + 10² + 11² + 12² + 13² + 14² = 1015. That makes 1015 the 14th square pyramidal number.

331 + 337 + 347 = 1015. That’s the sum of 3 consecutive prime numbers.

1015 is the hypotenuse of four Pythagorean triples:
119-1008-1015
168-1001-1015
609-812-1015
700-735-1015

1015 looks interesting when it is written is some other bases:
707 in BASE 12
1D1 in BASE 26 (D is 13 base 10)
TT in BASE 34 (T is 29 base 10)
T0 in BASE 35

  • 1015 is a composite number.
  • Prime factorization: 1015 = 5 × 7 × 29
  • 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 1015 has exactly 8 factors.
  • Factors of 1015: 1, 5, 7, 29, 35, 145, 203, 1015
  • Factor pairs: 1015 = 1 × 1015, 5 × 203, 7 × 145, or 29 × 35
  • 1015 has no square factors that allow its square root to be simplified. √1015 ≈ 31.85906

Level 2 and Simplifying √1014

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Today’s Puzzle:

Have you memorized a basic multiplication table? If you have, then you can solve this puzzle. The numbers being multiplied together aren’t where they are in a regular multiplication table, but you can still easily figure out where they need to go. There is only one solution. I bet you can find it!

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

Simplifying √1014:

If I wanted to find √1014, I would first check to see if it were divisible by 4 or by 9 because most numbers whose square roots can be simplified are divisible by 4 or by 9 or both.
1014 isn’t divisible by 4 because 14 isn’t divisible by 4.
It isn’t divisible by 9 because 1 + 0 + 1 + 4 = 6, and 6 is not divisible by 9.
However, it is divisible by both 2 and 3 and thus also by 6. Since most people are less likely to make a mistake dividing by 6 in ONE step instead of two, I would make a little division cake and do that division first:

Recognizing that 169 is a perfect square, I would then take the square root of everything on the outside of my little cake. (√6)(√169) = 13√6

Factors of 1014:

  • 1014 is a composite number.
  • Prime factorization: 1014 = 2 × 3 × 13 × 13, which can be written 1014 = 2 × 3 × 13²
  • The exponents in the prime factorization are 1, 1, and 2. Adding one to each and multiplying we get (1 + 1)(1 + 1)(2 + 1) = 2 × 2 × 3 = 12. Therefore 1014 has exactly 12 factors.
  • Factors of 1014: 1, 2, 3, 6, 13, 26, 39, 78, 169, 338, 507, 1014
  • Factor pairs: 1014 = 1 × 1014, 2 × 507, 3 × 338, 6 × 169, 13 × 78, or 26 × 39,
  • Taking the factor pair with the largest square number factor, we get √1014 = (√169)(√6) = 13√6 ≈ 31.84337

Sum-Difference Puzzles:

6 has two factor pairs. One of those pairs adds up to 5, and the other one subtracts to 5. Put the factors in the appropriate boxes in the first puzzle.

1014 has six factor pairs. One of the factor pairs adds up to ­65, and a different one subtracts to 65. If you can identify those factor pairs, then you can solve the second puzzle!

The second puzzle is really just the first puzzle in disguise. Why would I say that?

More about the Number 1014:

Because 13² is one of its factors, 1014 is the hypotenuse of two Pythagorean triples:
714-720-1014 which is 6 times (119-120-169),
390-936-1014 which is (5-12-13) times 78.

1014 looks interesting when written in some other bases:
It’s 600 in BASE 13 because 6(13²) = 6 (169) = 1014,
and 222 in BASE 22 because 2(22²) + 2(22) + 2(1) = 2(484 + 22 + 1) = 2(507) = 1014.