A Multiplication Based Logic Puzzle

Posts tagged ‘Multiplication table’

805 and Level 4

23 × 35 = 805 so we shouldn’t be surprised that 805 is palindrome NN in BASE 34. N is the same as 23 in base 10. Thus NN can be derived from 23(34) + 23(1) = 23(34 + 1) = 23 × 35 = 805. NN obviously is divisible by 11 like all 2 digit palindromes are.

Since 23 = 22 + 1, should we expect that 805 is a palindrome in BASE 22? No, and that is for the same reason that not all multiples of 11 are palindromes.

Finding the factors in today’s puzzle shouldn’t be very difficult, but the last few might be trickier than the rest:

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

  • 805 is a composite number.
  • Prime factorization: 805 = 5 x 7 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 805 has exactly 8 factors.
  • Factors of 805: 1, 5, 7, 23, 35, 115, 161, 805
  • Factor pairs: 805 = 1 x 805, 5 x 161, 7 x 115, or 23 x 35
  • 805 has no square factors that allow its square root to be simplified. √805 ≈ 28.37252

805 is the hypotenuse of a Pythagorean triple:

  • 483-644-805, which is 3-4-5 times 161

805 can be written as the sum of three squares four ways:

  • 25² + 12² + 6² = 805
  • 24² + 15² + 2² = 805
  • 20² + 18² + 9² = 805
  • 18² + 16² + 15² = 805

 

 

 

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803 From Top to Bottom

8 – 0 + 3 = 11, so 803 is divisible by 11.

  • 803 is a composite number.
  • Prime factorization: 803 = 11 x 73
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 803 has exactly 4 factors.
  • Factors of 803: 1, 11, 73, 803
  • Factor pairs: 803 = 1 x 803 or 11 x 73
  • 803 has no square factors that allow its square root to be simplified. √803 ≈ 3372546

You can solve today’s Level 3 puzzle by starting at the top of the first column, finding the factors of the clues and writing them in the appropriate cells. Then continue to go down that same column, cell by cell, finding factors and writing them down until you reach the bottom. Make sure that both the first column and the top row have each number from 1 to 10 written in them.

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

Here’s a few more facts about the number 803:

803 is the hypotenuse of a Pythagorean triple:

  • 528-605-803 which is 11 times another Pythagorean triple: 48-55-73

803 is the sum of three squares six different ways:

  • 27² + 7² + 5² = 803
  • 25² + 13² + 3² = 803
  • 23² + 15² + 7² = 803
  • 21² + 19² + 1² = 803
  • 19² + 19² + 9² = 803
  • 17² + 17² + 15² = 803

803 is the sum of consecutive prime numbers three different ways. Prime factor 11 is not in any of those ways, but prime factor 73 is in two of them.

  • 263 + 269 + 271 = 803, that’s 3 consecutive primes.
  • 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 = 803, that’s 9 consecutive primes.
  • 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 = 803, that’s 13 consecutive primes.

803 is a palindrome in two bases. Why are the numbers similar in these two palindromes?

  • 30203 BASE 4 because 3(256) + 0(64) + 2(16) + 0(4) + 3(1) = 803
  • 323 BASE 16 because 3(16²) + 2(16) + 3(1) = 803

 

 

 

802 Pi Day at Smith’s

In the United States tomorrow’s date is written 3-14. Because 3.14 is a famous approximation for π (pi), people all over the country will eat pie to celebrate Pi Day. This afternoon I took a picture of this sign and the pie display at my local Smith’s Food and Drug.

I took that picture right when I walked into the store, but there were no pies on display for National Pi Day.

About 15 minutes later I returned to the display to take another picture. Now there were pies on the table! I told a salesperson who I think worked on the display that I was going to take a picture and put it on my blog. She asked what kind of a blog I wrote. I told her a math blog. She looked puzzled and asked why I would want to put a picture of pies on a math blog. Then she turned around, looked at the display, and said something like, “Oh, now I get it, the number pi.”

How do you choose between apple, cherry, or peach pie? It’s much easier if you choose two and then you can get a free 8 oz. Cool Whip, too. Yummy.

If by chance you prefer pizza pi, here’s a thought from twitter that is often repeated in March:

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And here’s some original artwork that displays pi in a way I had never thought of before:

//platform.twitter.com/widgets.js

BREAKING: secret of Pi revealed #PiDay pic.twitter.com/Ao8BQp31jd

//platform.twitter.com/widgets.js

You can also look here for a million digits of pi.

But pi is not the only interesting number in the world. Every number has its own curiosities. Let me tell you some reasons to get excited about the number 802:

802 is the sum of two squares:

  • 21² + 19² = 802

So 802 is the hypotenuse of a Pythagorean triple:

  • 80-798-802, which is 2 times another triple: 40-399-401.

It also means something else: Since odd numbers 21 and 19 have no common prime factors, 802 can be evenly divide by 2. Duh. . ., but it also means that unless 802 is also divisible by 5, 13, or 17, its only factors will be 2 and a prime number! Why are those three numbers the only ones I care about? Because they are the only prime number Pythagorean triple hypotenuses less than √802 ≈ 28.3.

Guess what? 5, 13, and 17 do not divide evenly into 802, so 802 is the product of 2 and a prime number which happens to be 401.

  • 802 is a composite number.
  • Prime factorization: 802 = 2 x 401
  • The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 802 has exactly 4 factors.
  • Factors of 802: 1, 2, 401, 802
  • Factor pairs: 802 = 1 x 802 or 2 x 401
  • 802 has no square factors that allow its square root to be simplified. √802 ≈ 28.3196045

Today’s puzzle is number 802 to distinguish it from every other puzzle I’ve made. Writing the numbers 1 – 10 in both the top row and the first column so that the factors and the clues work together as a multiplication table is as easy as pie!

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

And here is a little more about the number 802:

802 is the sum of 8 consecutive prime numbers:

  • 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 = 802

802 can also be written as the sum of three squares three different ways:

  • 28² + 3² + 3² = 802
  • 27² + 8² + 3² = 802
  • 24² + 15² + 1² = 802

802 is also a palindrome in two other bases:

  • 414 BASE 14 because 4(196) + 1(14) + 4(1) = 802
  • 202 BASE 20 because 2(400) + 0(20) + 2(1) = 802

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).

 

 

 

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

710 and Level 1

  • 710 is a composite number.
  • Prime factorization: 710 = 2 x 5 x 71
  • 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 710 has exactly 8 factors.
  • Factors of 710: 1, 2, 5, 10, 71, 142, 355, 710
  • Factor pairs: 710 = 1 x 710, 2 x 355, 5 x 142, or 10 x 71
  • 710 has no square factors that allow its square root to be simplified. √710 ≈ 26.645825.

Here is today’s puzzle:

 

710 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-14

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Here are few more facts about 710:

710 can be written as the sum of consecutive numbers several ways:

  • 176 + 177 + 178 + 179 = 710; that’s four consecutive numbers.
  • 140 + 141 + 142 + 143 + 144 = 710; that’s five consecutive numbers.
  • 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 = 710; that’s 20 consecutive numbers.

710 is the sum of the twenty prime numbers from 3 to 73.

Because 5 is one of its factors, 710 is the hypotenuse of Pythagorean triple 426-568-710. What is the greatest common factor of those three numbers?

710 is palindrome 868 in BASE 9; note that 8(81) + 6(9) + 8(1) = 710.

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710 Factors

705 and Level 2

  • 705 is a composite number.
  • Prime factorization: 705 = 3 x 5 x 47
  • 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 705 has exactly 8 factors.
  • Factors of 705: 1, 3, 5, 15, 47, 141, 235, 705
  • Factor pairs: 705 = 1 x 705, 3 x 235, 5 x 141, or 15 x 47
  • 705 has no square factors that allow its square root to be simplified. √705 ≈ 26.551836.

Here is today’s puzzle:

 

705 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-12-07

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What else can I tell you about the number 705?

Because 5 is one of its factors, 705 is the hypotenuse of the Pythagorean triple 423-564-705. What is the greatest common factor of those three numbers?

705 is the sum of consecutive numbers several different ways:

  • 352 + 353 = 705; (2 consecutive numbers)
  • 234 + 235 + 236 = 705; (3 consecutive numbers)
  • 139 + 140 + 141 + 142 + 143 = 705; (5 consecutive numbers)
  • 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 = 705; (15 consecutive numbers)

705 is palindrome 1A1 in base 22; note that A is equivalent to 10 base 10, 22² = 484, and 1(484) + 10(22) + 1(1) = 705.

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705 Factors

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