A Multiplication Based Logic Puzzle

Posts tagged ‘factor trees’

882 Factor Trees for the First Day of Autumn

September 22 was the first day of autumn. Leaves are already beginning to fall from the trees.

To rake up the leaves for 882, you might first notice that it’s even. The logical thing to do would be to first divide 882 by two. . . But perhaps you might notice that 8 + 8 + 2 = 18, a number divisible by nine, so you might just as logically begin by dividing 882 by 9. The first step you take determines how the factor tree looks.

882 has many possible factor trees but these two are probably the most common.

You can rake the leaves up this way or you can rake them up that way, but when you rake up the leaves from 882’s factor trees, you always get the same prime factors: 2, 3, 3, 7, and 7.

Here’s a little more about the number 882:

882 has eighteen factors. The greatest number less than 882 with eighteen factors is 828. Now get this: 288 also has eighteen factors. That means that every possible combination of 8-8-2 has exactly eighteen factors!

882 has some interesting representations in some other bases:

  • 616 BASE 12, because 6(12²) + 1(12)¹ + 6(12º) = 882
  • 242 BASE 20, because 2(20²) + 4(20)¹ + 2(20º) = 882
  • 200 BASE 21, because 2(21²) = 882

882 is also the sum of consecutive primes: 439 + 443 = 882

  • 882 is a composite number.
  • Prime factorization: 882 = 2 × 3 × 3 × 7 × 7, which can be written 882 = 2 × 3² × 7²
  • The exponents in the prime factorization are 1, 2 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1)(2 + 1) = 2 × 3 × 3 = 18. Therefore 882 has exactly 18 factors.
  • Factors of 882: 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 49, 63, 98, 126, 147, 294, 441, 882
  • Factor pairs: 882 = 1 × 882, 2 × 441, 3 × 294, 6 × 147, 7 × 126, 9 × 98, 14 × 63, 18 × 49 or 21 × 42
  • Taking the factor pair with the largest square number factor, we get √882 = (√441)(√2) = 21√2 ≈ 29.6984848.

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How to Find Consecutive Even or Odd Numbers That Add up to 840

840 has more factors than any previous number. Those factors will help us write 840 as the sum of consecutive counting numbers, consecutive even numbers, and consecutive odd numbers. What are the factors of 840? Here are a couple of the many possible factor trees for 840:

The red leaves on the tree are prime numbers. Gathering the six red leaves from either factor tree above gives us 840’s prime factorization: 840 = 2³ × 3 × 5 × 7.  Now 840 is not the smallest number to have six red leaves. In fact, there are smaller numbers with as many as nine leaves, but 840’s six innocent-looking red leaves will turn into a whopping 32 factors!

  • 840 is a composite number.
  • Prime factorization: 840 = 2 × 2 × 2 × 3 × 5 × 7, which can be written 840 = 2³ × 3 × 5 × 7.
  • The exponents in the prime factorization are 3, 1, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) )(1 + 1) = 4 × 2 × 2 × 2 = 32. Therefore 840 has exactly 32 factors.
  • Factors of 840: 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840
  • Factor pairs: 840 = 1 × 840, 2 × 420, 3 × 280, 4 × 210, 5 × 168, 6 × 140, 7 × 120, 8 × 105, 10 × 84, 12 × 70, 14 × 60, 15 × 56, 20 × 42, 21 × 40, 24 × 35 or 28 × 30
  • Taking the factor pair with the largest square number factor, we get √840 = (√4)(√210) = 2√210 ≈ 28.98275.

840 is also the smallest number that can be evenly divided by the first eight counting numbers!

How to find consecutive counting numbers that add up to 840:

840 is more than the 40th triangular number (820) and less than the 41st triangular number (861). We can also arrive at the number 40 by using √(1 + 840×2) – 1 = 40, no rounding necessary. 840 has six odd factors that are not more than 40, namely 1, 3, 5, 7, 15, 21 and 35.

I’ll describe the ways we can write 840 as the sum of consecutive numbers. Can you write out the sums? I’ve done one of them for you:

  • using 3 numbers with 280 as the middle number,
  • using 5 numbers with 168 as the middle number,
  • using 7 numbers with 120 as the middle number; 117 + 118 + 119 + 120 + 121 + 122 + 123 = 840
  • using 15 numbers with 56 as the middle number,
  • using 21 numbers with 40 as the middle number
  • using 35 numbers with 24 as the middle number.

Notice each of those ways has a factor pair of 840 in the description.

The largest power of 2 that is a factor of 840 is 8, which doubled becomes 16. Which of 840’s odd factors multiplied by 16 are not more than 40? 1 × 16 = 16, and 3 × 16 = 48. Oops, that’s too much. The rest of its odd factors times 16 will be too much as well. Nevertheless, we can write 840 as the sum of 16 counting numbers. 840÷16 = 52.5 so 52 and 53 will be the 8th and 9th numbers in the sum.

  • 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 = 840

Only even numbers can be the sum of consecutive even numbers. How do we find consecutive EVEN numbers that add up to an even number? Let’s use 840 as an example again. 840÷2 = 420.

First we will find all the ways to write 420 as the sum of consecutive numbers. Then we will simply double the middle number and surround it with the appropriate number of even numbers to get a sum of even numbers that add up to 840:

√(1 + 420×2) – 1 = 28, no rounding necessary, so we will make a list of the odd factors that are not more than 28. They are 1, 3, 5, 7, 15, and 21. We also note that the largest power of 2 that is a factor of 420 is 4. Doubling 4, we get 8. Which of 420’s odd factors multiplied by 8 are not more than 28? 1 × 8 = 8, and 3 × 8 = 24. All of the rest will be too much.

I’ll describe the ways we can write 840 as the sum of consecutive even numbers. You can see 420’s factor pairs and 840’s factor pairs in the descriptions. Can you write out the sums? I’ve done one of them for you:

  • using 3 even numbers with 2 × 140 = 280 as the middle number
  • using 5 even numbers with 2 × 84 = 168 as the middle number
  • using 7 even numbers with 2 × 60 = 120 as the middle number
  • using 15 even numbers with 2 × 28 = 56 as the middle number
  • using 21 even numbers with 2 × 20 = 40 as the middle number; 20 + 22 + 24 + 26 + 28 + 30 + 32 + 34 + 36 + 38 + 40 + 42 + 44 + 46 + 48 + 50 + 52 + 54 + 56 + 58 + 60 = 840

We can also write 840 as the sum of an even amount of consecutive even numbers.

  • using 8 even numbers: 2 times (49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 = 420) becomes 98 + 100 + 102 + 104 + 106 + 108 + 110 + 112 = 840. Notice that 840÷8 = 105, the odd number that is between the two numbers in the exact middle of the sum.
  • Likewise, using 24 even numbers: 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 + 30 + 32 + 34 + 36 + 38 + 40 + 42 + 44 + 46 + 48 + 50 + 52 + 54 + 56 + 58 = 840. Notice that 840÷ 24 = 35, the odd number that is between the two numbers in the exact middle of the sum.

How to find consecutive ODD numbers that add up to an even number like 841: 

Let me first point out that every square number, n², is the sum of the first n numbers.

For example: 100 = 10², and 100 is also the sum of the first 10 odd numbers as illustrated below:

A similar statement is true for every other square number:

Yes, every square number, n², is the sum of the first n odd numbers.

To write a non-square even number as the sum of consecutive odd numbers, the first thing we must do is determine if the number can be written as the difference of two squares. If an even number has a factor pair, b × a, in which b > a, and BOTH a and b are even, then that even number can be expressed as the difference of two squares by using [(b + a)/2]² – [(b – a)/2]².

Now as long as an even number can be expressed as the difference of two squares, , then that number can be written as the sum of all the odd numbers from 2A+1 to 2B-1.

840 is an even number with eight factor pairs in which both numbers are even. Let’s use those even factor pairs to find the ways to write 840 as the sum of consecutive ODD numbers:

  • 420 × 2 = 840 means 211² 209² = 840, and that means the sum of the 2 consecutive odd numbers from 419 to 421 = 840
  • 210 × 4 = 840 means 107² 103² = 840, and that means the sum of the 4 consecutive odd numbers from 207 to 213 = 840
  • 140 × 6 = 840 means 73² 67² = 840, and that means the sum of the 6 consecutive odd numbers from 135 to 145 = 840
  • 84 × 10 = 840 means 47² 37² = 840, and that means the sum of the 10 consecutive odd numbers from 75 to 93 = 840
  • 70 × 12 = 840 means 41² 29² = 840, and that means the sum of the 12 consecutive odd numbers from 59 to 81 = 840
  • 60 × 14 = 840 means 37² 23² = 840, and that means the sum of the 14 consecutive odd numbers from 47 to 73 = 840
  • 42 × 20 = 840 means 31² 11² = 840, and that means the sum of the 20 consecutive odd numbers from 23 to 61 = 840
  • 30 × 28 = 840 means 29² = 840, and that means the sum of the 28 consecutive odd numbers from 3 to 29 = 840

Thus, 840 with its record setting 32 factors, can be written as the sum of 7 consecutive numbers, 7 consecutive even numbers, and 8 consecutive odd numbers!

Incidentally, being able to write 840 as the difference of two squares, eight different ways also makes 840 a leg in at least eight different Pythagorean triples. Those Pythagorean triples can be calculated using the numbers from the difference of two squares. For example, 682-840-1082 can be calculated from 2(31)(11), 31² 11², 31² + 11².

840 was the leg for those eight triples. It is possible that looking at 2(b)(a), where b × a = 420, will produce some more Pythagorean triples with 840 as the leg.

840 is also the hypotenuse of one Pythagorean triple, 504-672-840.

On a note totally unrelated to anything I’ve written above, 840 is a repdigit in two bases:

  • SS BASE 29 (S is 28 base 10) Note that 28(29) + 28(1) = 28(30) = 840
  • 00 BASE 34 (0 is 24 base 10) Note that 24(34) + 24(1) = 24(35) = 840

840 is also the sum of twin prime numbers 419 and 421.

780 is the 39th Triangular Number

  • 780 is a composite number.
  • Prime factorization: 780 = 2 x 2 x 3 x 5 x 13, which can be written 780 = (2^2) x 3 x 5 x 13
  • 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 x 3 x 2 x 2 = 24. Therefore 780 has exactly 24 factors.
  • Factors of 780: 1, 2, 3, 4, 5, 6, 10, 12, 13, 15, 20, 26, 30, 39, 52, 60, 65, 78, 130, 156, 195, 260, 390, 780
  • Factor pairs: 780 = 1 x 780, 2 x 390, 3 x 260, 4 x 195, 5 x 156, 6 x 130, 10 x 78, 12 x 65, 13 x 60, 15 x 52, 20 x 39, or 26 x 30
  • Taking the factor pair with the largest square number factor, we get √780 = (√4)(√195) = 2√195 ≈ 27.92848.

780-factor-pairs

There are MANY ways to make factor trees for 780. Here are just three of them:

780 Factor Trees

Stetson.edu informs us that (7 + 5)(8 + 5)(0 + 5) = 780.

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

  • 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 = 780 (10 consecutive primes).
  • 191 + 193 + 197 + 199 = 780 (4 of my favorite consecutive primes).

26 and 30 are both exactly 2 numbers away from their average 28, so 780 can be written as the difference of two squares:

780 = 26 x 30 = (28 – 2)(28 + 2) = 28² – 2² = 784 – 4.

(It can be written as the difference of two squares three other ways, but I won’t list them here.)

Because 5 and 13 are both factors of 780, it is the hypotenuse of FOUR Pythagorean triangles making each of these equations true:

  • 192² + 756² = 780²
  • 300² + 720² = 780²
  • 396² + 672² = 780²
  • 468² + 624² = 780²

780 can be written as the sum of three perfect squares two different ways:

  • 26² + 10² + 2² = 780
  • 22² + 14² + 10² = 780

780 BASE 10 is palindrome QQ BASE 29. (Q is 26 in BASE 10.) Note that 26(29) + 26(1) = 780.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 = 780, making 780 the 39th triangular number. Since 20 is in the exact middle of that list of numbers, 780 is also the 20th hexagonal number.

Triangular numbers are interesting, but are they good for anything? Here’s one good thing:

Count the Terms of Sums Squared

There are 26 letters in the English alphabet. 39 variables would use the alphabet exactly one and one half times, but that’s okay because we can use upper case letters the second time around:

(a + b + c + . . . + x + y+ z + A + B + C + . . . + K + L + M)² has exactly 780 terms because it has 39 single variables, and 780 is the 39th triangular number.

780 is the 39th triangular number because 39⋅40/2 = 780.

A couple of months ago I saw a fascinating image on twitter. It is a square made with 1001 dots representing the numbers from -500 to +500 with zero in the exact middle. The triangular numbers are represented by the “+” pattern seen here. If we added about 600 more dots to the square, -780 and +780 would also lie on that +.

//platform.twitter.com/widgets.js

Half of all triangular numbers are also hexagonal numbers. If there were enough dots, then -780 and +780 would also be among the dots forming the “-” pattern representing the hexagonal numbers. 780 is the 20th hexagonal number because 20(2⋅20-1) = 780.

 

672 Is it too soon to pick out a tree?

672 can make MANY factor trees.  Here I’ve pictured only a few of the possibilities, one for each of its factor pairs (excluding 1 x 672).

Is it too soon to pick out a tree?

672 Factor Trees

Every one of those trees has the prime factors of 672: 2, 2, 2, 2, 2, 3, and 7, but finding them on each tree might be a challenge because I didn’t distinguish the prime factors from the other factors. Some of those prime factors might seem like they are lost in a pile of leaves. Can you find them on each tree?

672 is the 9th number to have 24 factors. Here is a number line highlighting all nine of those numbers and the distances between them.

24 Factors Number Line

Notice the difference between 672 and the previous number with 24 factors is 12, a record low.

You might get the impression looking at the number line that numbers having 24 factors might be much more common from now on. That may be true, nevertheless, the next number after 672 to have exactly 24 factors is 756 which is 84 numbers away and well past 720 the smallest number to have 30 factors.

Indeed infinitely many numbers have 24 factors, but probably 672 is the last one that will get much attention.

The numbers in the factor pair 24 and 28 are each exactly two numbers away from 26, their average. That means we are just 2² numbers away from 26².

In other words, 672 equals 26² – 2² which can be factored into (26 + 2)(26 – 2) so (26 + 2)(26 – 2) = 28 x 24 = 672.

  • 672 is a composite number.
  • Prime factorization: 672 = 2 x 2 x 2 x 2 x 2 x 3 x 7, which can be written 672 = (2^5) x 3 x 7
  • The exponents in the prime factorization are 5, 1, and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1)(1 + 1) = 6 x 2 x 2 = 24. Therefore 672 has exactly 24 factors.
  • Factors of 672: 1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 32, 42, 48, 56, 84, 96, 112, 168, 224, 336, 672
  • Factor pairs: 672 = 1 x 672, 2 x 336, 3 x 224, 4 x 168, 6 x 112, 7 x 96, 8 x 84, 12 x 56, 14 x 48, 16 x 42, 21 x 32, or 24 x 28
  • Taking the factor pair with the largest square number factor, we get √672 = (√16)(√42) = 4√42 ≈ 25.92296.

650 is the sum of all the clues in this Level 1 puzzle

1² + 2² + 3² + 4² + 5² + 6² +7² + 8² + 9² +10² + 11² + 12²  = 650

Thus 650 is the 12th square pyramidal number and can be calculated using 12(12 +1)(2⋅12 + 1)/6.

If you add up all the clues in today’s Find the Factors puzzle, you will get the number 650. However, if you print the puzzle from the excel file, one of the clues is missing because it isn’t needed to find the solution.

650 is the hypotenuse of seven Pythagorean triples!

  • 72-646-650
  • 160-630-650
  • 182-624-650
  • 250-600-650
  • 330-560-650
  • 390-520-650
  • 408-506-650

Can you find the greatest common factor of each triple? Each greatest common factor will be one of the factors of 650 listed below the puzzle.

650 is the hypotenuse of so many Pythagorean triples because it is divisible by 5, 13, 25, 65, and 325 which are also hypotenuses of triples. The smallest three numbers to be the hypotenuses of at least 7 triples are 325, 425, and 650.

Since 25 x 26 = 650, we know that (25-1)(26 + 1) = 650 – 2. Thus 24 x 27 = 648.

650 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-19

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  • 650 is a composite number.
  • Prime factorization: 650 = 2 x 5 x 5 x 13, which can be written 650 = 2 x (5^2) x 13
  • 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 650 has exactly 12 factors.
  • Factors of 650: 1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650
  • Factor pairs: 650 = 1 x 650, 2 x 325, 5 x 130, 10 x 65, 13 x 50, or 25 x 26
  • Taking the factor pair with the largest square number factor, we get √650 = (√25)(√26) = 5√26 ≈ 25.495098.

650 Trees

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640 Fall Factor Trees and Level 5

Because it is fall, and 640 has many factors, I decided to make factor trees using fall colors. Get out your rakes!

640 factor trees

There are many other possible factor trees for 640, but raking leaves can be a lot of work, so I only made two of them.

640 is the hypotenuse of the Pythagorean triple 384-512-640.

Stetson.edu informs us that 640 = 16!!!!!!, but if you type 16!!!!!! into a calculator, you will get an error message as soon as you type !!.

16!!!!!! ≠ (((((16!)!)!)!)!)!

There are 6 !’s so 16!!!!!! = 16(16-6)(16-12) = 16 x 10 x 4.

Here is this week’s Level 5 puzzle:

640 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-05

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  • 640 is a composite number.
  • Prime factorization: 640 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5, which can be written 640 = (2^7) x 5
  • The exponents in the prime factorization are 7 and 1. Adding one to each and multiplying we get (7 + 1)(1 + 1) = 8 x 2 = 16. Therefore 640 has exactly 16 factors.
  • Factors of 640: 1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 320, 640
  • Factor pairs: 640 = 1 x 640, 2 x 320, 4 x 160, 5 x 128, 8 x 80, 10 x 64, 16 x 40, or 20 x 32
  • Taking the factor pair with the largest square number factor, we get √640 = (√64)(√10) = 8√10 ≈ 25.298221

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640 Logic

630 Factor Trees and Level 2

630 is the 7th number with exactly 24 factors. So far, the seven numbers counting numbers with 24 factors are 360, 420, 480, 504, 540, 600, and 630. No counting number less than 630 has more than 24 factors.

Two of those seven numbers make up the Pythagorean triple 378-504-630. Which factor of 630 is the greatest common factor of those three numbers in the triple?

Here are a few of the MANY possible factor trees for 630.

630 Factor Trees

630 is the sum of the six prime numbers from 97 to 113.

630 is the 35th triangular number because (35 x 36)/2 = 630. It is also the 18th hexagonal number because 18(2 x 18 – 1) = 630.

630 is a triangular number that is a multiple of other triangular numbers in more ways than you probably want to know:

  • 630 is three times the 20th triangular number, 210, because 3(20 x 21)/2 = 630.
  • 630 is 6 times the 14th triangular number, 105, because 6(14 x 15)/2 = 630.
  • 630 is 14 times the 9th triangular number, 45, because 14(9 x 10)/2 = 630.
  • 630 is 30 times the 6th triangular number, 21, because 30(6 x 7)/2 = 630.
  • 630 is 42 times the 5th triangular number, 15, because 42(5 x 6)/2 = 630.
  • 630 is 63 times the 4th triangular number, 10, because 63(4 x 5)/2 = 630.
  • 630 is 105 times the 3rd triangular number, 6, because 105(3 x 4)/2 = 630.
  • 630 is 210 times the 2nd triangular number, 3, because 210(2 x 3)/2 = 630.
  • and finally, 630 is 630 times the 1st triangular number, 1, because 630(1 x 2)/2 = 630

630 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-28

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  • 630 is a composite number.
  • Prime factorization: 630 = 2 x 3 x 3 x 5 x 7, which can be written 630 = 2 x (3^2) x 5 x 7
  • The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 x 3 x 2 x 2 = 24. Therefore 630 has exactly 24 factors.
  • Factors of 630: 1, 2, 3, 5, 6, 7, 9, 10, 14, 15, 18, 21, 30, 35, 42, 45, 63, 70, 90, 105, 126, 210, 315, 630
  • Factor pairs: 630 = 1 x 630, 2 x 315, 3 x 210, 5 x 126, 6 x 105, 7 x 90, 9 x 70, 10 x 63, 14 x 45, 15 x 42, 18 x 35, or 21 x 30
  • Taking the factor pair with the largest square number factor, we get √630 = (√9)(√70) = 3√70 ≈ 25.09980.

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630 Factors

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