1296 Last Digit Effort

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If you always use a calculator to find the roots of perfect powers like 1296, then you will likely miss out on discovering the wonderful patterns those powers have. Look at the table below. What do you notice? What do you wonder?

When I looked at it, I noticed several patterns and I wondered about a few things that I explored to see if they were true. However, I won’t tell you what I noticed or what conjectures I made until the end of this post, because I don’t want to spoil YOUR chance to discover those patterns yourself. I will tell you that there are MANY patterns in the table above and that most of my conjectures were true, but one of them turned out not to be.

Besides the patterns that you will have to discover for yourself, the table can be very useful: You can use the table to find several fractional powers of the numbers in the body of the table. For example, if you wanted to find out what is 1296^(7/4), then you could simply run your finger down the n⁷ column and the n⁴ column simultaneously until you reach 1296 on the n⁴ column (because 4 is the denominator of the fractional power). Your answer, 279936, will be in the n⁷ column on the same row as 1296.

However, if you wanted to find 1296^(3/2), you would either have to expand the table to include 36² and 36³, or you would have to use the fact that 3/2 = 6/4. Then you could use the process of the previous paragraph to see that
1296^(3/2) = 1296^(6/4) = 46656.

Now I’ll share some more facts about the number 1296:

  • 1296 is a composite number, a perfect square, and a perfect fourth power.
  • Prime factorization: 1296 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3, which can be written 1296 = 2⁴ × 3⁴
  • The exponents in the prime factorization are 4 and 4. Adding one to each and multiplying we get (4 + 1)(4 + 1) = 5 × 5 = 25. Therefore 1296 has exactly 25 factors.
  • Factors of 1296: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 81, 108, 144, 162, 216, 324, 432, 648, 1296
  • Factor pairs: 1296 = 1 × 1296, 2 × 648, 3 × 432, 4 × 324, 6 × 216, 8 × 162, 9 × 144, 12 × 108, 16 × 81, 18 × 72, 24 × 54, 27 × 48, or 36 × 36
  • Taking the factor pair with the largest square number factor, we get √1296 = (√36)(√36) = 36.

1296 is the smallest number with exactly 25 factors.

There are MANY different factor trees you could make for 1296, but here are some made with squares:

1296 is the 21st Friedman number because of its digits and 9²×16 = 1296. (The same digits are on both sides of that equation.)

Since 36 is a triangular number and 36² = 1296, we get this wonderful, powerful fact:
(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)² = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ = 1296

1296 looks interesting and often rather square when it is written in several bases:
It’s1210000 in BASE 3,
10000 in BASE 6,
2420 in BASE 8,
1700 in BASE 9,
1296 in BASE 10
900 in BASE 12,
789 in BASE 13 (Ha, ha. Seven ate nine!),
484 in BASE 17,
400 in BASE 18,
169 in BASE 33,
144 in BASE 34,
121 in BASE 35, and
100 in BASE 36

Do you know the square roots in base 10 of all the numbers in bold? Indeed, 1296 is a fascinating square number!

Here are some things I noticed on the table of powers for numbers ending in . . .

  • 1: The last digit is always 1
  • 2: The pattern for the last digit is 2, 4, 8, 6, repeating
  • 3: The pattern for the last digit is 3, 9, 7, 1, repeating
  • 4: The pattern for the last digit is 4, 6, 4, 6, repeating
  • 5: The last digit is always 5
  • 6: The last digit is always 6
  • 7: The pattern for the last digit is 7, 9, 3, 1, repeating (3’s pattern backward but starting with 7)
  • 8: The pattern for the last digit is 8, 4, 2, 6, repeating (2’s pattern backward but starting with 8)
  • 9: The pattern for the last digit is 9, 1, 9, 1, repeating
  • 0: The last digit is always 0

For the squares:

  • 1² and 9² both end in 1, and 1 + 9 = 10
  • 2² and 8² both end in 4, and 2 + 8 = 10
  • 3² and 7² both end in 9, and 3 + 7 = 10
  • 4² and 6² both end in 6, and 4 + 6 = 10

For the cubes:

  • 1³ ends in 1, and 9³ ends in 9
  • 2³ ends in 8, and 8³ ends in 2; 2 + 8 = 10
  • 3³ ends in 7, and 7³ ends in 3; 3 + 7 = 10
  • 4³ ends in 4, and 6³ ends in 6

Other observations:

  • No matter what n is, the last digit of n, n⁵, and n⁹ are always the same. Yes, that pattern continues with n¹³ and so forth.
  • The number of digits for 10ᵃ is a+1, and the number of digits for 9ᵃ is a.

Dan Bach noticed something else.

I had never heard of Benford’s Law before, so I’ve learned something new here!  All of us can learn something new by observing patterns and listening to each other.

1295 Mystery Level

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This pinwheel-shaped puzzle is no little kid’s toy. The logic needed to solve it might be a mystery to see, but it is still there!

Print the puzzles or type the solution in this excel file: 12 factors 1289-1299

Here are some facts about the number 1295:

  • 1295 is a composite number.
  • Prime factorization: 1295 = 5 × 7 × 37
  • 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 1295 has exactly 8 factors.
  • Factors of 1295: 1, 5, 7, 35, 37, 185, 259, 1295
  • Factor pairs: 1295 = 1 × 1295, 5 × 259, 7 × 185, or 35 × 37
  • 1295 has no square factors that allow its square root to be simplified. √1295 ≈ 35.98611

Since 35 × 37 = 1295, we know that 1295 is one number less than 36².

1295 is the hypotenuse of FOUR Pythagorean triples:
399-1232-1295 which is 7 times (57-176-185)
420-1225-1295 which is 35 times (12-35-37)
728-1071-1295 which is 7 times (104-153-185)
777-1036-1295 which is (3-4-5) times 259

1294 and Level 6

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You will need to use multiplication facts from 1 to 12, logic, and perseverance to solve this tricky level 6 puzzle. Good luck!

Print the puzzles or type the solution in this excel file: 12 factors 1289-1299

Here are some facts about the number 1294:

  • 1294 is a composite number.
  • Prime factorization: 1294 = 2 × 647
  • 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 1294 has exactly 4 factors.
  • Factors of 1294: 1, 2, 647, 1294
  • Factor pairs: 1294 = 1 × 1294 or 2 × 647
  • 1294 has no square factors that allow its square root to be simplified. √1294 ≈ 35.97221

1294 is also the sum of the twenty prime numbers from 23 to 107.

1293 and Level 5

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Only factors from 1 to 12 are allowed in the factor pairs of the clues in these puzzles. Find a row or column with only one allowable common factor to start this puzzle. As you progress, factors for other rows or columns will be eliminated. Keep at it, and you will succeed!

Print the puzzles or type the solution in this excel file: 12 factors 1289-1299

Now I’ll share a few facts about the number 1293:

  • 1293 is a composite number.
  • Prime factorization: 1293 = 3 × 431
  • 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 1293 has exactly 4 factors.
  • Factors of 1293: 1, 3, 431, 1293
  • Factor pairs: 1293 = 1 × 1293 or 3 × 431
  • 1293 has no square factors that allow its square root to be simplified. √1293 ≈ 35.95831

1293 is palindrone 141 in BASE 34 because 34² + 4(34) + 1 = 1293

1292 and Level 4

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Finding the Factors in this puzzle will require you to use logic and you will get a bonus: your knowledge of the multiplication table will be a little better than it was before.

Print the puzzles or type the solution in this excel file: 12 factors 1289-1299

Now I’ll give you some information about today’s puzzle number:

  • 1292 is a composite number.
  • Prime factorization: 1292 = 2 × 2 × 17 × 19, which can be written 1292 = 2² × 17 × 19
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1292 has exactly 12 factors.
  • Factors of 1292: 1, 2, 4, 17, 19, 34, 38, 68, 76, 323, 646, 1292
  • Factor pairs: 1292 = 1 × 1292, 2 × 646, 4 × 323, 17 × 76, 19 × 68, or 34 × 38
  • Taking the factor pair with the largest square number factor, we get √1292 = (√4)(√323) = 2√323 ≈ 36.9444

1292 is the hypotenuse of a Pythagorean triple:
608-1140-1292 which is (8-15-17) times 76

1291 and Level 3

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If you think of the common factors of 25 and 55, then you have actually started to solve this puzzle! Start at the top of the puzzle and work your way down. Try it!

Print the puzzles or type the solution in this excel file: 12 factors 1289-1299

Here are some facts about the number 1291

  • 1291 is a prime number.
  • Prime factorization: 1291 is prime.
  • The exponent of prime number 1291 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1291 has exactly 2 factors.
  • Factors of 1291: 1, 1291
  • Factor pairs: 1291 = 1 × 1291
  • 1291 has no square factors that allow its square root to be simplified. √1291 ≈ 35.93049

How do we know that 1291 is a prime number? If 1291 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1291 ≈ 35.9. Since 1291 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1291 is a prime number.

1291 is also palindrome 1D1 in BASE 30 because 30² + 13(30) + 1 = 1291

 

1290 Multiplication Boomerang

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Do multiplication and division facts seem like something you threw out long ago but still come back to hit you? Perhaps this puzzle can help you get more familiar with those facts so they won’t hurt you so much anymore.

Print the puzzles or type the solution in this excel file: 12 factors 1289-1299

That was puzzle number 1290. Here are some facts about that number:

  • 1290 is a composite number.
  • Prime factorization: 1290 = 2 × 3 × 5 × 43
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1290 has exactly 16 factors.
  • Factors of 1290: 1, 2, 3, 5, 6, 10, 15, 30, 43, 86, 129, 215, 258, 430, 645, 1290
  • Factor pairs: 1290 = 1 × 1290, 2 × 645, 3 × 430, 5 × 258, 6 × 215, 10 × 129, 15 × 86, or 30 × 43
  • 1290 has no square factors that allow its square root to be simplified. √1290 ≈ 35.91657

1290 is the sum of two consecutive prime numbers:
643 + 647 = 1290

1290 is the hypotenuse of a Pythagorean triple:
774-1032-1290 which is (3-4-5) times 258

1289 and Level 1

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You might think this is a very easy puzzle, but for some people, it will be challenging, and will hopefully help them learn some multiplication facts better.

Print the puzzles or type the solution in this excel file: 12 factors 1289-1299

Since this is puzzle number 1289, I’ll share some facts about that number:

  • 1289 is a prime number.
  • Prime factorization: 1289 is prime.
  • The exponent of prime number 1289 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1289 has exactly 2 factors.
  • Factors of 1289: 1, 1289
  • Factor pairs: 1289 = 1 × 1289
  • 1289 has no square factors that allow its square root to be simplified. √1289 ≈ 35.90265

How do we know that 1289 is a prime number? If 1289 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1289 ≈ 35.9. Since 1289 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1289 is a prime number.

1289 is the sum of two squares:
35² + 8² = 1289

1289 is the hypotenuse of a Pythagorean triple:
560-1161-1289 calculated from 2(35)(8), 35² – 8², 35² + 8²

Here’s another way we know that 1289 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 35² + 8² = 1289 with 35 and 8 having no common prime factors, 1289 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1289 ≈ 35.9. Since 1289 is not divisible by 5, 13, 17, or 29, we know that 1289 is a prime number.

1288 Mystery Puzzle

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How difficult is this mystery level puzzle? That is part of the mystery, but I assure you that if you use logic and basic multiplication facts you can find the unique solution.

Print the puzzles or type the solution in this excel file: 10-factors-1281-1288

That was puzzle #1288. Now I’ll tell you some facts about the number 1288.

  • 1288 is a composite number.
  • Prime factorization: 1288 = 2 × 2 × 2 × 7 × 23, which can be written 1288 = 2³ × 7 × 23
  • 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 1288 has exactly 16 factors.
  • Factors of 1288: 1, 2, 4, 7, 8, 14, 23, 28, 46, 56, 92, 161, 184, 322, 644, 1288
  • Factor pairs: 1288 = 1 × 1288, 2 × 644, 4 × 322, 7 × 184, 8 × 161, 14 × 92, 23 × 56, or 28 × 46
  • Taking the factor pair with the largest square number factor, we get √1288 = (√4)(√322) = 2√322 ≈ 37.88872

1288 has four factor pairs that contain only even factors so 1288 can be written as the difference of two squares four different ways:
323² – 321² = 1288
163² – 159² = 1288
53² – 39² = 1288
37² – 9² = 1288

 

1287 Mystery Level

/ ivasallay / Leave a comment

I’ve put twenty clues in this mystery level puzzle. Some of the factors will be easy to find, but some of them won’t be quite as easy. Use logic the entire time, and you will be able to solve it!

Print the puzzles or type the solution in this excel file: 10-factors-1281-1288

Now I’ll share some facts about the number 1287:

  • 1287 is a composite number.
  • Prime factorization: 1287 = 3 × 3 × 11 × 13, which can be written 1287 = 3² × 11 × 13
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1287 has exactly 12 factors.
  • Factors of 1287: 1, 3, 9, 11, 13, 33, 39, 99, 117, 143, 429, 1287
  • Factor pairs: 1287 = 1 × 1287, 3 × 429, 9 × 143, 11 × 117, 13 × 99, or 33 × 39
  • Taking the factor pair with the largest square number factor, we get √1287 = (√9)(√143) = 3√143 ≈ 35.87478

1287 is the hypotenuse of a Pythagorean triple:
495-1188-1287 which is (5-12-13) times 99