A Multiplication Based Logic Puzzle

Archive for the ‘Level 1 Puzzle’ Category

801 and Level 1

When it comes to applying our tried and true trick for divisibility by nine to the number 801, zero is just a place holder. Thus, since 81 is divisible by 9, so is 801. Adding up its digits was hardly necessary.

  • 801 is a composite number.
  • Prime factorization: 801 = 3 x 3 x 89, which can be written 801 = (3^2) x 89
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 801 has exactly 6 factors.
  • Factors of 801: 1, 3, 9, 89, 267, 801
  • Factor pairs: 801 = 1 x 801, 3 x 267, or 9 x 89
  • Taking the factor pair with the largest square number factor, we get √801 = (√9)(√89) = 3√89 ≈ 28.301943396.

Would you be surprised to know the following division facts?

  • 81 ÷ 3 = 27
  • 801 ÷ 3 = 267
  • 8001 ÷ 3 = 2667
  • 80001 ÷ 3 = 26667 and so forth. The number of 6’s in the quotient is the same as the number of 0’s in the dividend!

Here are some more predictable division facts:

  • 81 ÷ 9 = 9
  • 801 ÷ 9 = 89
  • 8001 ÷ 9 = 889
  • 80001 ÷ 9 = 8889 and so forth. You guessed it! The number of 8’s in the quotient is the same as the number of 0’s in the dividend!

Even though you can’t see 81 in this puzzle with all perfect square clues, it isn’t difficult to see where 9 × 9 and 81 belong:

Print the puzzles or type the solution on this excel file: 10-factors 801-806

801 is a palindrome in three bases:

  • 1441 BASE 8 because 1(8^3) + 4(8^2) + 4(8) + 1(1) = 801
  • 2D2 BASE 17 D is 13 base 10 because 2(289) + 13(17) = 2(1) = 801
  • 171 BASE 25 because 1(25²) + 7(25) + 1(1) =801

801 is the sum of two squares:

  • 24² + 15² =801

So it follows that 801 is the hypotenuse of a Pythagorean triple:

  • 351-720-801 which is 9 times 39-80-89

801 is the sum of three squares TEN ways:

  1. 28² + 4² + 1² = 801
  2. 27² + 6² + 6² = 801
  3. 26² + 11² + 2² =801
  4. 26² + 10² + 5² = 801
  5. 24² + 12² + 9² = 801
  6. 23² + 16² + 4² = 801
  7. 22² + 14² + 11² = 801
  8. 21² + 18² + 6² = 801
  9. 20² + 20² + 1² = 801
  10. 17² + 16² + 16² = 801

Stetson.edu gives us this last fun fact:

801 = (7! + 8! + 9! + 10!) / (7 × 8 × 9 × 10).

 

 

 

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795 a Level 1 Puzzle with ♥

I learned yesterday that pushing ALT on the computer key pad and 3 on the number pad at the same time produces a when the keys are released. I added a few ‘s to today’s puzzle because Valentine’s day is almost here:

795-puzzle

  • 795 is a composite number.
  • Prime factorization: 795 = 3 x 5 x 53
  • 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 x 2 x 2 = 8. Therefore 795 has exactly 8 factors.
  • Factors of 795: 1, 3, 5, 15, 53, 159, 265, 795
  • Factor pairs: 795 = 1 x 795, 3 x 265, 5 x 159, or 15 x 53
  • 795 has no square factors that allow its square root to be simplified. √795 ≈ 28.195744.

795-factor-pairs

Stetson.edu informs us that if we add up all the factors of 795 we get a number that is a perfect 4th power:

  • 1 + 3 + 5 + 15 + 53 + 159 + 265 + 795 = 1296, which happens to be the 4th power of 6. (Of course 1296 also equals 36 squared.)

795 is the hypotenuse of four Pythagorean triples and thus:

  • 69 – 792 – 795 which is 3 times (23 – 264 – 265)
  • 288 – 741 – 795 which is 3 times (96 – 247 – 265)
  • 420 – 675 – 795 which is 15 times (28 – 45 – 53)
  • 477 – 636 – 795 which is 159 times (3 – 4 – 5)

795 is also the sum of three squares two different ways:

  • 25² + 13² + 1² = 795
  • 25² + 11² + 7² = 795

 

 

788 and Level 1

Since 88, its last two digits, are divisible by 4, we know that 788 and every other whole number ending in 88 is divisible by 4.

I learned the following fascinating fact about these six numbers starting with 788 from Stetson.edu:

788-consecutive-numbers

788 is also palindrome 404 in BASE 14. Note that 4(196) + 0(14) + 4(1) = 788.

788 is the hypotenuse of Pythagorean triple 112-780-788 which is 28-195-197 times 4.

788-puzzle

Print the puzzles or type the solution on this excel file: 10-factors-788-794

  • 788 is a composite number.
  • Prime factorization: 788 = 2 x 2 x 197, which can be written 788 = (2^2) x 197
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 788 has exactly 6 factors.
  • Factors of 788: 1, 2, 4, 197, 394, 788
  • Factor pairs: 788 = 1 x 788, 2 x 394, or 4 x 197
  • Taking the factor pair with the largest square number factor, we get √788 = (√4)(√197) = 2√197 ≈ 28.071338.

788-factor-pairs

782 and Level 1

  • 782 is a composite number.
  • Prime factorization: 782 = 2 x 17 x 23
  • 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 x 2 x 2 = 8. Therefore 782 has exactly 8 factors.
  • Factors of 782: 1, 2, 17, 23, 34, 46, 391, 782
  • Factor pairs: 782 = 1 x 782, 2 x 391, 17 x 46, or 23 x 34
  • 782 has no square factors that allow its square root to be simplified. √782 ≈ 27.96426.

782-factor-pairs

Here’s today’s puzzle:

782-puzzle

 Print the puzzles or type the solution on this excel file: 12-factors-782-787

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Here’s some more information about the number 782:

Stetson.edu informs us that the sum of the factors of 782 is not only a perfect square number but also a perfect 4th power.

Specifically, 1 + 2 + 17 + 23 + 34 + 46 + 391 + 782 = 1296, which happens to be 6^4.

23 is one of the factors of 782, and here are three interesting number facts about the number 782 that involve the number 23:

  • 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 = 782. (That’s 23 consecutive numbers beginning with the number 23.)
  • 782 is the 23rd pentagonal number because 23(3⋅23 – 1)/2 = 782.
  • 782 is palindrome NN in BASE 33. (N is 23 base 10.) Note that 23(33) + 23(1) = 782

Because 17 is one of its factors, 782 is the hypotenuse of Pythagorean triple 368-690-782.

Thus, 368² + 690² = 782².

Also since 17 x 46 = 782, the greatest common factor of 368, 690 and 782 is 46.

782 is also the sum of three squares six different ways:

  • 27² + 7² + 2² = 782
  • 26² + 9² + 5² = 782
  • 25² + 11² + 6² = 782
  • 22² + 17² + 3² = 782
  • 19² + 15² + 14² = 782
  • 18² + 17² + 13² = 782

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782-factors

774 and Level 1

  • 774 is a composite number.
  • Prime factorization: 774 = 2 x 3 x 3 x 43, which can be written 774 = 2 x (3^2) x 43
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 x 3 x 2 = 12. Therefore 774 has exactly 12 factors.
  • Factors of 774: 1, 2, 3, 6, 9, 18, 43, 86, 129, 258, 387, 774
  • Factor pairs: 774 = 1 x 774, 2 x 387, 3 x 258, 6 x 129, 9 x 86, or 18 x 43
  • Taking the factor pair with the largest square number factor, we get √774 = (√9)(√86) = 3√86 ≈ 27.820855.

774-factor-pairs

Here is a factoring puzzle that is just right for beginners:

774 Puzzle

Print the puzzles or type the solution on this excel file: 10-factors-2016

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Here’s a little more about the number 774:

774 is the sum of consecutive prime numbers two different ways:

  • 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 = 774; that’s 18 consecutive primes.
  • 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 = 774; that’s 14 consecutive primes.

774 is the sum of three triangular numbers eight different ways:

  • 666 + 105 + 3 = 774
  • 630 + 78 + 66 = 774
  • 561 + 210 + 3 = 774
  • 528 + 231 + 15 = 774
  • 528 + 210 + 36 = 774
  • 465 + 231 + 78 = 774
  • 378 + 351 + 45 = 774
  • 378 + 276 + 120 = 774

774 is also the sum of three squares nine different ways:

  • 27² + 6² + 3² = 774
  • 26² + 7² + 7² = 774
  • 25² + 10² + 7² = 774
  • 23² + 14² + 7² = 774
  • 22² + 17² + 1² = 774
  • 22² + 13² + 11² = 774
  • 21² + 18² + 3² = 774
  • 18² + 15² + 15² = 774
  • 17² + 17² + 14² = 774

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774 Factors


 

 

768 and Level 1

768 is made from three consecutive numbers so it is divisible by 3.

  • 768 is a composite number.
  • Prime factorization: 768 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3, which can be written 768 = (2^8) x 3
  • The exponents in the prime factorization are 8 and 1. Adding one to each and multiplying we get (8 + 1)(1 + 1) = 9 x 2 = 18. Therefore 768 has exactly 18 factors.
  • Factors of 768: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 768
  • Factor pairs: 768 = 1 x 768, 2 x 384, 3 x 256, 4 x 192, 6 x 128, 8 x 96, 12 x 64, 16 x 48 or 24 x 32
  • Taking the factor pair with the largest square number factor, we get √768 = (√256)(√3) = 16√3 ≈ 27.7128129.

768-factor-pairs

Today’s puzzle is relatively easy:

768 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-02-25

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Here’s a few more facts about the number 768:

255 + 256 + 257 = 768. That is the ONLY way to write 768 as the sum of consecutive positive whole numbers.

There is also only one way to write 768 as the sum of three square numbers:

  • 16² + 16² + 16² = 768.

768 is the sum of eight consecutive prime numbers: 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 = 768.

768 is 1100000000 in BASE 2, 30000 in BASE 4, and 300 in BASE 16.

768 is also a palindrome in a few bases:

  • 363 BASE 15; note that 3(225) + 6(15) + 3(1) = 768
  • OO BASE 31 (O = 24 base 10); note that 24(31) + 24(1) = 768
  • CC BASE 63 (C = 12 base 10); note that 12(63) + 12(1) = 768
  • 66 BASE 127; note that 6(127) + 6(1) = 768
  • 33 BASE 255; note that 3(255) + 3(1) = 768

What do all those BASES have in common? They are all one less than a power of 2.

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768 Factors

761 and Level 1

  • 761 is a prime number.
  • Prime factorization: 761 is prime and cannot be factored.
  • The exponent of prime number 761 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 761 has exactly 2 factors.
  • Factors of 761: 1, 761
  • Factor pairs: 761 = 1 x 761
  • 761 has no square factors that allow its square root to be simplified. √761 ≈ 27.586228.

761-factor-pairs

How do we know that 761 is a prime number? If 761 were not a prime number, then it would be divisible by at least one prime number less than or equal to √761 ≈ 27.6. Since 761 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 761 is a prime number.

There will be a lot about squares in today’s post, and all of the clues in today’s puzzle are perfect squares, too:

761 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-02-04

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Here’s more about the number 761:

Because 761 is a prime number, there is only one way to write it as the sum of consecutive numbers: 380 + 381 = 761.

However, because it is a centered square number, 761 can be written as the sum of consecutive square numbers: 20² + 19² = 761.

761 Centered Square Number

And because it is the sum of two consecutive square numbers, it is also the hypotenuse of a primitive Pythagorean triple: 39-760-761 which was calculated using 20² – 19², 2(20)(19), 20² + 19².

Notice also that 20 + 19 = 39, ⌊39²/2⌋ = 760, and ⌈39²/2⌉ = 761. (Those peculiar parenthesis mean⌊round down to the closest whole number⌋ and ⌈round up to the closest whole number⌉.)

In the 39 x 39 graphic shown above there are 760 white squares and 761 squares with some color. Thus, looking at that graphic is one way to look at the primitive Pythagorean triple 39-760-761.

All of that gives us another way to know that 761 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 20² + 19² = 761 with 20 and 19 having no common prime factors, 761 will be prime unless it is divisible by a prime number Pythagorean hypotenuse less than or equal to √761 ≈ 27.6. Since 761 is not divisible by 5, 13, or 17, we know that 761 is a prime number.

761 can also be expressed as the sum of three squares TEN ways:

  • 27² + 4² + 4² = 761
  • 26² + 9² + 2² = 761
  • 26² + 7² + 6² = 761
  • 25² + 10² + 6² = 761
  • 24² + 13² + 4² = 761
  • 24² + 11² + 8² = 761
  • 23² + 14² + 6² = 761
  • 22² + 14² + 9² = 761
  • 21² + 16² + 8² = 761
  • 19² + 16² + 12² = 761

761 is a palindrome in two different bases:

  • 535 BASE 12; note that 5(144) + 3(12) + 5(1) = 761.
  • 1I1 BASE 20 (I = 18 base 10); note that 1(400) + 18(20) + 1(1) = 761.

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761 Factors

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