A Multiplication Based Logic Puzzle

Archive for the ‘How to Find the Factors of a Number’ Category

960 Factor Trees

960 is the smallest number to have exactly 28 factors. 960 is 2⁶·3·5, so any factor tree made for it will have 6 + 1 + 1 = 8 prime factors. Since 8 is a power of 2, this number, 960, has some beautiful and well-balanced factor trees as well as some that aren’t as good-looking. Here are five of the MANY possible factor trees for 960:

 

960 can be written as the difference of 2 squares TEN different ways:

  1. 241² – 239² = (241 + 239)(241 – 239) = 480 × 2 = 960
  2. 122² – 118² = (122 + 118)(122 – 118) = 240 × 4 = 960
  3. 83² – 77² = (83 + 77)(83 – 77) = 160 × 6 = 960
  4. 64² – 56² = (64 + 56)(64 – 56) = 120 × 8 = 960
  5. 53² – 43² = (53 + 43)(53 – 43) = 96 × 10 = 960
  6. 46² – 34² = (46 + 34)(46 – 34) = 80 × 12 = 960
  7. 38² – 22² = (38 + 22)(38 – 22) = 60 × 16 = 960
  8. 34² – 14² = (34 + 14)(34 – 14) = 48 × 20 = 960
  9. 32² – 8² = (32 + 8)(32 – 8) = 40 × 24 = 960
  10. 31² – 1² = (31 + 1)(31 – 1) = 32 × 30 = 960

960 is the sum of the sixteen prime numbers from 29 to 97.

It is also the sum of six consecutive prime numbers:
149 + 151 + 157 + 163 + 167 + 173 = 960

960 is the hypotenuse of a Pythagorean triple:
576-768-960 which is (3-4-5) times 192

I like how 960 looks in these other bases:
33000 in BASE 4 because 3(4⁴) + 3(4³) = 3(256 + 64) = 3 × 320 = 960
440 in BASE 15 because 4(15²) + 4(15) = 4(225 + 15) = 4 × 240 = 960
UU in BASE 31 (U is 30 base 10), because 30(31) + 30(1) = 30(31 + 1) = 30 × 32 = 960
U0 in BASE 32 because 30(32) + 0 = 960

Stetson.edu informs us that 9 + 6 + 09³ + 6³ + 0³ = 960

  • 960 is a composite number and a perfect square.
  • Prime factorization: 960 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 5, which can be written 960 = 2⁶ × 3 × 5
  • The exponents in the prime factorization are 6, 1 and 1. Adding one to each and multiplying we get (6 + 1)(1 + 1)(1 + 1) = 7 × 2 × 2 = 28. Therefore 960 has exactly 28 factors.
  • Factors of 960: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, 80, 96, 120, 160, 192, 240, 320, 480, 960
  • Factor pairs: 960 = 1 × 960, 2 × 480, 3 × 320, 4 × 240, 5 × 192, 6 × 160, 8 × 120, 10 × 96, 12 × 80, 15 × 64, 16 × 60, 20 × 48, 24 × 40, or 30 × 32
  • Taking the factor pair with the largest square number factor, we get √960 = (√64)(√15) = 8√15 ≈ 30.9838668

Advertisements

Applying Divisibility Tricks to 924

Let’s apply some basic divisibility tricks to find some of the factors of 924:

  1. Like every other counting number, 924 is divisible by 1.
  2. Since 924 is even, it is divisible by 2.
  3. 9 + 2 + 4 = 15, a number divisible by 3, so 924 is divisible by 3.
  4. Its last two digits, 24, is divisible by 4, so 924 is divisible by 4.
  5. Its last digit isn’t 0 or 5, so 924 is NOT divisible by 5.
  6. 924 is even and divisible by 3, so it is also divisible by 6.
  7. Since 92-2(4) = 84, a number divisible by 7, we know that 924 is also divisible by 7.
  8. Since its last two digits are divisible by 8, and the third to the last digit, 9, is odd, 924 is NOT divisible 8.
  9. 9 + 2 + 4 = 15, a number not divisible by 9, so 924 is NOT divisible by 9.
  10. The last digit is not 0, so 924 is NOT divisible by 10.
  11. 9 – 2 + 4 = 11, so 924 is divisible by 11.

Thus, 1, 2, 3, 4, 6, 7, and 11 are all factors of 924. You can see its prime factors easily on the outside of its prime factor cake:

You may have seen one of its many possible factor trees contained in the first frame of this factor tree for 852,852 from my previous post:

924 is in the very center of the 12th row of Pascal’s triangle because 12!/(6!6!) = 924.

924 is the sum of consecutive prime numbers: 461 + 463 = 924

I like 924 written in some other bases:

770 BASE 11

336 BASE 17

220 BASE 21

SS BASE 32, S is 28

S0 BASE 33

  • 924 is a composite number.
  • Prime factorization: 924 = 2 x 2 x 3 x 7 x 11, which can be written 924 = 2² x 3 x 7 x 11
  • 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) = 3 x 2 x 2 x 2 = 24. Therefore 924 has exactly 24 factors.
  • Factors of 924: 1, 2, 3, 4, 6, 7, 11, 12, 14, 21, 22, 28, 33, 42, 44, 66, 77, 84, 132, 154, 231, 308, 462, 924
  • Factor pairs: 924 = 1 x 924, 2 x 462, 3 x 308, 4 x 231, 6 x 154, 7 x 132, 11 x 84, 12 x 77, 14 x 66, 21 x 44, 22 x 42, or 28 x 33,
  • Taking the factor pair with the largest square number factor, we get √924 = (√4)(√231) = 2√231 ≈ 30.3973683.

924 has several sets of consecutive factors. Besides being divisible by the 1st, 2nd, and 3rd triangular numbers (1, 3, and 6), those consecutive factors mean the following:

  • 924 is divisible by the 6th triangular number, 21, which is 6(7)/2.
  • 924 is divisible by the 11th triangular number, 66, which is 11(12)/2.
  • 924 is divisible by the 21st triangular number, 231, which is 21(22)/2.

660 My Two-Year Blogiversary

Today is my 2 year blogiversary and my 660th post. Thank you to all my readers and followers. I really appreciate all of you.

What better way is there to celebrate than with a gorgeous cake? Enjoy!

The cake method of finding prime factors makes a beautiful cake for the number 660 because 660 has several prime factors and the largest one, 11, looks like a couple of candles to top it off perfectly.

660 cake

660 has a lot of factors, 24 in fact. 660 is a special number for several reasons:

No number less than 660 has more factors than it does, but 360, 420, 480, 504, 540, 600, and 630 each have just as many.

660 has so many factors that it seems natural for it to between twin primes, 659 and 661.

660 is the hypotenuse of the Pythagorean triple 396-528-660. What is the greatest common factor of those three numbers?

660 is the sum of consecutive prime numbers three different ways. Prime number 101 is in two of those ways:

  • 157 + 163 + 167 + 173 = 660
  • 101 + 103 + 107 + 109 + 113 + 127 = 660
  • 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 = 660

—————————————————————————————————

  • 660 is a composite number.
  • Prime factorization: 660 = 2 x 2 x 3 x 5 x 11, which can be written 660 = 2² x 3 x 5 x 11
  • 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) = 3 x 2 x 2 x 2 = 24. Therefore 660 has exactly 24 factors.
  • Factors of 660: 1, 2, 3, 4, 5, 6, 10, 11, 12, 15, 20, 22, 30, 33, 44, 55, 60, 66, 110, 132, 165, 220, 330, 660
  • Factor pairs: 660 = 1 x 660, 2 x 330, 3 x 220, 4 x 165, 5 x 132, 6 x 110, 10 x 66, 11 x 60, 12 x 55, 15 x 44, 20 x 33, or 22 x 30
  • Taking the factor pair with the largest square number factor, we get √660 = (√4)(√165) = 2√165 ≈ 25.690465.

—————————————————————————————————

 

 

420 Factor Trees

Did you know that the sum of all the prime numbers between 100 and 110 equals 420? Yes, 101 + 103 + 107 + 109 = 420.

Since 20 × 21 = 420, we know that 420 is the sum of the first 20 EVEN numbers. Thus,

  • 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 + 30 + 32 + 34 + 36 + 38 + 40 = 420.

420 is the smallest number that can be divided evenly by all the natural numbers from 1 to 7.

420 has a LOT of factors, more than most people would think it does.  In fact, of all of the numbers from 1 to 420, there is only one number, 360, that has as many factors as 420 has.

There are 4 different prime numbers that can divide evenly into 420. Here are those factor trees:

420 prime number factor trees

Many, but not all, of the factors of 420 are listed somewhere on those four trees.

  • 420 is a composite number.
  • Prime factorization: 420 = 2 x 2 x 3 x 5 x 7, which can be written 420 = (2^2) x 3 x 5 x 7
  • 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) = 3 x 2 x 2 x 2 = 24. Therefore 420 has exactly 24 factors.
  • Factors of 420: 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420
  • Factor pairs: 420 = 1 x 420, 2 x 210, 3 x 140, 4 x 105, 5 x 84, 6 x 70, 7 x 60, 10 x 42, 12 x 35, 14 x 30, 15 x 28, or 20 x 21
  • Taking the factor pair with the largest square number factor, we get √420 = (√4)(√105) = 2√105 ≈ 20.4939

Each factor pair, except 1 x 420, can make its own factor tree. Here are some factor trees featuring the other seven factor pairs:

420 other factor trees

If I hadn’t made all the prime numbers in red, it’s possible that one or more of the prime numbers might get forgotten. That is why I prefer the cake method for finding the prime factorization of a number. All of the prime numbers are listed in numerical order on the outside of the cake.

Finding prime factors of 420

 

120 and Level 5

120  is a composite number. 120 = 1 x 120, 2 x 60, 3 x 40, 4 x 30, 5 x 24, 6 x 20, 8 x 15, or 10 x 12. Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120. Prime factorization: 120 = 2 x 2 x 2 x 3 x 5, which can also be written 120 = 2³ x 3 x 5.

Thinking process using divisibility tricks to find the factor pairs of 120:

√120 is irrational and approximately equal to 10.95. Every factor pair of 120 will have one factor less than 10.95 and one factor greater than 10.95, and we will find both factors in each pair at the same time. The following numbers are less than 10.95. Are they factors of 120?

  1. Yes, all whole numbers are divisible by 1, so 1 x 120 = 120.
  2. Yes, 120 is an even number. 120 ÷ 2 = 60, so 2 x 60 = 120. (Since 60 is even, 4 will also be a factor of 120.)
  3. Yes, 1 + 2 + 0 = 3 which is divisible by 3 (but not by 9), so 120 is divisible by 3. 120 ÷ 3 = 40, so 3 x 40 = 120. Note 120 will not be divisible by 9.
  4. Yes, the number formed from its last two digits, 20, is divisible by 4, so 120 is divisible by 4, and 4 x 30 = 120. (Note since 30 is even, 8 will also be a factor of 120.)
  5. Yes, the last digit is 0 or 5, so 120 is divisible by 5, and 5 x 24 = 120.
  6. Yes, 120 is divisible by both 2 and 3, so it is divisible by 6, and 6 x 20 = 120.
  7. No. The divisibility trick for 7 requires us to split 120 into 12 and 0. We double 0 and subtract the double from 12. 12 – (2 x 0) = 12 – 0 = 12. Since 12 is not divisible by 7, 120 also is not divisible by 7.
  8. Yes, see 4 above. 120 = 8 x 15. (This will mean that ANY number whose last 3 digits are 120 will also be divisible by 8.)
  9. No, see 3 above. 120 is not divisible by 9.
  10. Yes, 120 ends with a zero, so 10 is a factor of 120, and 10 x 12 = 120.

From this thinking process we conclude that the factor pairs of 120 are 1 x 120, 2 x 60, 3 x 40, 4 x 30, 5 x 24, 6 x 20, 8 x 15, and 10 x 12.

When 120 is a clue in the FIND THE FACTORS 1 – 12 puzzles, use 10 and 12 as the factors.

5! = 1 x 2 x 3 x 4 x 5 = 120.

2014-19 Level 5

Excel file of this week’s puzzles and last week’s factors: 10 Factors 2014-05-12

2014-19 Level 5 Logic

33 and Some Divisibility Tricks for 3 and 9

33 is a composite number. 33 = 1 x 33 or 3 x 11. Factors of 33: 1, 3, 11, 33. Prime factorization: 33 = 3 x 11.

When 33 is a clue in the FIND THE FACTORS 1 – 12 puzzles, always use 3 and 11 as the factors.

———————————————————————————-

What patterns do you see in the following chart?

multiples of 9 chart

Probably you noticed or a teacher taught you the easy way to remember what 9 times a numbers from one to ten is just as the chart illustrates.

Did you also ever notice that if you add up the digits of the multiples of 9 in the multiplication table, you get 9?

What is really great is if you add up the digits of ANY multiple of 9, you’ll get 9 or some other multiple of 9! This is called a divisibility trick because it is a way to find out if a number is divisible by 9 without actually dividing by 9.

The same trick works on multiples of 3: If you add up the digits of a multiple of 3, you will get 3 or some other multiple of 3! Lets apply these divisibility tricks to a few numbers:

Is 243 divisible by 3 or 9? We don’t have to divide to know the answer:

2 + 4 + 3 = 9, which is divisible by both 3 and 9, so, yes, 243 is divisible by both 3 and 9.

If you do the actual division:

  • 243 ÷ 3 = 81
  • 243 ÷ 9 = 27.

———————————————————————————————-

Is 582 divisible by 3 or by 9? Add up the digits to find out: 5 + 8 + 2 = 15.

Since 15 is divisible by 3, but not by 9, we know 582 is divisible by 3, but not by 9.

If you do the actual division:

  • 582 ÷ 3 = 194
  • 582 ÷ 9 = 64 Remainder 6.

When we added the digits of 582, we got 15. Notice that 1 + 5 = 6, the remainder when we divided 582 by 9.

When you add up the digits of a number until you have only one digit left, if that digit is not 9, then that number is the remainder you would get if you did the actual division!

———————————————————————————————-

Now let’s see if 1753 is divisible by 3 or 9.

1 + 7 + 5 + 3 = 16; 1 + 6 = 7.

7 cannot be evenly divided by 3 or 9, so 1753 is not divisible by 3 or 9.

If you do the actual division:

  • 1753 ÷ 9 = 194 Remainder 7 (the same 7 that equals 1 + 6 above).
  • 1753 ÷ 3 = 584 Remainder 1

Notice that 7 ÷ 3 = 2 Remainder 1

———————————————————————————————-

All of these problems demonstrate that if you add the digits of a number until you are left with a single digit, if that digit is 3, 6, or 9, then the original number is divisible by 3.

The last problem demonstrates that if you divide that single digit by 3, the remainder will be the same if you divided the original number by 3.

These divisibility tricks for 3 and 9 can give quite a bit of valuable information!

 

21 Factors of the Year 2013 and 2014

21 is a composite number. 21 = 1 x 21 or 3 x 7. Factors of 21: 1, 3, 7, 21. Prime factorization: 21 = 3 x 7.

When 21 is a clue in the FIND THE FACTORS puzzles, use 3 x 7.

Scroll down the page to find factoring information about 2013 and 2014.

2013 year

Near the end of each year movie critics make lists of the ten best movies and the ten worse movies of the year. News agencies list the ten most significant news stories. Time magazine lists the ten most influential people of the year. The music industry lists the top ten songs of the year. As 2013 draws to a close, it is most appropriate for me to review the factors of the year.

2013 had exactly 8 positive factors. These factors were 1, 3, 11, 33, 61, 183, 671, and 2013.

There is no room for argument. I am absolutely certain this list is complete. No one will make any comments disagreeing with me, calling me names, or asking how I could have left Two or Five or Seven off the list. Also no one will wonder why I would include forgettable 671 on the list. Do the Math. 671 was clearly a factor in 2013. Three of the factors of 2013 were also prime factors. They were 3, 11, and 61. This graphic clearly shows those prime factors.

2013 tree

2013 also had 8 negative factors. The first negative factor on the list is no surprise: Minus One. Year in and year out we can count on Minus One being a negative factor. Some other factors were just as negative in 2013, namely -3, -11, -33, – 61, -183, -671, and -2013. Of course, many of those factors were so obscure that most people never gave them a second thought all year long. Again I expect no arguments or negative comments on these selections. Anyone who knows anything about factors will have to agree with this list.

Even though 2014 hasn’t even started, I am going to predict the factors of 2014, and I am absolutely positive that my predictions will be 100% correct. You will not even have to wait until the end of 2014 to verify my accuracy.

The positive factors of 2014 will be (drum roll) 1, 2, 19, 38, 53, 106,1007, and 2014.

Most people expect the number One to be a positive factor every single year, and it will not let us down in 2014. The number Two has a reputation of being a factor only about half the time. Since she was not a factor at all in 2013, I am confident that she will get her act together again in 2014 and become a factor once more. All the other factors I’ve listed have not been factors for a very long time, and each one of them is due to make a difference over and over again in 2014 until they have nothing leftover. I predict that 2014 will have three prime factors, namely 2, 19, and 53, as illustrated in the following graphic.

2014 tree

How can I make such accurate assessments and spot on predictions? I will tell you: I work with factors almost every single day, and I’ve spent years observing them. Every time I have been given an assignment to become acquainted with them, I have approached that assignment with enthusiasm and determination.

Regardless of my astounding record, YOU can become just as much an expert as I am with just a little bit of knowledge and effort. You may discover, as I have, that factoring can be great fun. Here are a couple of logic puzzles that require factoring to solve: 

2013-12-30.22013-12-30.3

All you have to do to solve one of the puzzles is write the numbers 1 – 12 in the top row and again in the first column so that those numbers are the factors of the given clues. Each puzzle has only one solution.

At the top of this post is a page titled How to Find the Factors, and it gives hints to solve the puzzles.   Click 12 Factors 2013-12-30 to find a printable version of these and a few other puzzles as well as the solutions for last week’s puzzles. Excel or comparable spreadsheet program is needed to open the file.

Have a great 2014 and happy factoring!

Related Articles

Tag Cloud