# 630 Factor Trees and Level 2

### Today’s Puzzle:

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

### Factor Trees for 630:

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.

### Factors of 630:

• 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.

### Sum-Difference Puzzle:

630 has twelve factor pairs. One of the factor pairs adds up to 53, and a different one subtracts to 53. If you can identify those factor pairs, then you can solve this puzzle!

### More about the Number 630:

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.

# 540 and Level 4

### Today’s Puzzle:

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-29

Here is a logical order to use the clues to solve the puzzle:

### Factors of 540:

• 540 is a composite number.
• Prime factorization: 540 = 2 x 2 x 3 x 3 x 3 x 5, which can be written 540 = (2^2) x (3^3) x 5
• The exponents in the prime factorization are 2, 3 and 1. Adding one to each and multiplying we get (2 + 1)(3 + 1)(1 + 1) = 3 x 4 x 2 = 24. Therefore 540 has exactly 24 factors.
• Factors of 540: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, 270, 540
• Factor pairs: 540 = 1 x 540, 2 x 270, 3 x 180, 4 x 135, 5 x 108, 6 x 90, 9 x 60, 10 x 54, 12 x 45, 15 x 36, 18 x 30 or 20 x 27
• Taking the factor pair with the largest square number factor, we get √540 = (√36)(√15) = 6√15 ≈ 23.237900077

### Sum-Difference Puzzles:

60 has six factor pairs. One of those pairs adds up to 17, and  another one subtracts to 17. Put the factors in the appropriate boxes in the first puzzle.

540 has twelve factor pairs. One of the factor pairs adds up to 51, and a different one subtracts to 51. If you can identify those factor pairs, then you can solve the second puzzle!

The second puzzle is really just the first puzzle in disguise. Why would I say that?

### More about the Number 540:

540 is the sum of the fourteen consecutive prime numbers from 13 to 67. Can you list all those prime numbers? It is also the sum of consecutive prime numbers 269 and 271.

540 has the same number of factors as 504. Both of those numbers tie with 360, 420, and 480 for the most factors so far.

540 is the hypotenuse of the Pythagorean triple 324-432-540. What is the greatest common factor of those three numbers?

540 is also an untouchable number.

The sum of the interior angles of every convex pentagon total 540 degrees.

# 330 Christmas Factor Trees

### Today’s Puzzle:

Can you find the factors and complete this Christmas tree multiplication table?

Print the puzzles or type the factors on this excel file:  10 Factors 2014-12-22

### Factor Trees for 330:

Within these seven factor trees for 330 there are also factor trees for 6, 10, 15, 22, 30, 33, 55, 66, 110, and 165, the tops of which are all in brown. The prime factors of 330 are all in red.

### Factors of 330:

• 330 is a composite number.
• Prime factorization: 330 = 2 x 3 x 5 x 11
• 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 x 2 x 2 x 2 = 16. Therefore 330 has exactly 16 factors.
• Factors of 330: 1, 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, 165, 330
• Factor pairs: 330 = 1 x 330, 2 x 165, 3 x 110, 5 x 66, 6 x 55, 10 x 33, 11 x 30, or 15 x 22
• 330 has no square factors that allow its square root to be simplified. √330 ≈ 18.166

### Sum-Difference Puzzle:

330 has eight factor pairs. The numbers in one of those pairs add up to 61, and the numbers in another one subtract to 61. If you can identify those factors, then you can solve this puzzle!

# 24 Think You Have a Snow Problem? Try This:

### Snow Related Puzzles:

It’s January, and some of us are a little tired of the snow while others are longing to hit the ski slopes. I’m writing today to suggest a few ways to deal with snow when it seems like more of an obstacle than an opportunity.

I walked into a second-grade classroom last week and saw books and books about snow, snowflakes, snowmen, etc. Celebrating winter is one way to deal with the snow. Here are a few snow problems for you to solve while you review multiplication facts: I’ve created a nightmare of a snowstorm with huge snowflakes and snowballs and only an itsy bitsy snow shovel to deal with it all. Be grateful this isn’t what you are really facing! How do you solve one of these snow-related puzzles? 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.  Click here for some tips to help you solve the puzzles. Click 12 Factors 2014-01-13 for printable versions of these and a couple other puzzles as well as last week’s answers.

### Tools to Deal with Actual Snow:

Puzzles are fun, but they can’t help you get the car out of the garage after a snowstorm.

If you wake up to find just under a foot of snow (or even much more) on your driveway (or your roof), a good snow shovel can still be the answer. I recommend a shovel with a crooked handle. It makes lifting snow so much easier. This excellent article has everything you need to know to shovel snow.

I have not used cooking oil on my snow shovels, but I have sprayed it with non-stick cooking spray, and that did help when the snow was sticky. (Snow isn’t always sticky.)

Two places I have lived face winter storms on occasion. Those places were Oklahoma County, Oklahoma and Salt Lake County, Utah.

It didn’t snow very often when we lived near Oklahoma City, but when it did, a crooked-handled shovel allowed me to clear my driveway faster than any of my neighbors who had only a straight-handled shovel. Freezing rain, thunderstorms, and tornadoes were a much bigger concern in Oklahoma than snow.

Compared to Oklahoma, we get much more snow near Salt Lake City where we have lived for the last 20 years.   Now my family has two of these crooked handled shovels. For seventeen years we relied solely on those shovels. We found that shoveling was good exercise, but it also took too much time, and time seems to be getting in shorter supply.

Three years ago we bought a good quality snow blower, and it is definitely a time saver. If you buy a snow blower, make sure it is a model that allows the operator to change the direction where the snow is blown. Also if the snowblower gets clogged with snow, always turn it off before attempting to unclog it. We have a friend who lost a finger because he didn’t turn it off first.

We also know people who own tractor snowplows. They easily clear their driveways and sometimes the sidewalks for their entire block. There is a snowplow on the Sears website. The blade on it is currently for sale for just under \$250.00. Ouch.

We dream of someday getting an automatic snowmelt system that will turn itself on and melt any snow that falls on our driveway. One such system is described here.

Yes, there are many places that regularly get pounded with much more snow than Salt Lake City does. I’m not sure how helpful any of the above information will be for people who live in such places, so for anyone suffering in those locations I offer a few diversions:

### A Few Diversions from Dealing with the Snow:

My daughter-in-law endured the polar vortex that hit the Eastern half of the United States last week. She wrote, “Public service announcement: 37+ weeks pregnant with a head cold and two small sick kids is no fun. I do not recommend it. Today I am grateful for PBS kid shows, giant boxes of tissues, soft blankets, and easy slow cooker meals.” She also survived by finding humor in the situation. She shared a hilarious Sunny Street cartoon panel featuring a very cold pregnant woman whose water broke and turned immediately into ice cubes.

If you like making greeting cards, you could make a snowflake card using an early iteration of Koch’s snowflake. If you completed a few more iterations, you would discover that this snowflake is a lovely six-sided shape that happens to have a finite area, but an infinite perimeter. Here are the directions: http://renegadegeek.wordpress.com/2013/12/31/koch-snowflake-card/

If none of these tips or diversions help you solve or forget your snow problem, I have one more thing for you to try as a last resort:

I am absolutely amazed at how much more snow the Northern Hemisphere gets compared to the Southern Hemisphere during their respective winters. So as a last resort, maybe it is time to move.

I like this animation showing where it snows on earth each month of the year.

If you stay where you are, you may take comfort in knowing there are places that have much more snow. Check out the Top 12 Biggest Snowfall Events in Recorded History!

Twitter informs us that Alaska can deal with a lot of snow with some unusual snowplows:

### Factors of 24:

1. 24 is a composite number.
2. Prime factorization: 24 = 2 x 2 x 2 x 3, which can be written 24 = 2³ x 3
3. The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 24 has exactly 8        factors.
4. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
5. Factor pairs: 24 = 1 x 24, 2 x 12, 3 x 8, or 4 x 6
6. Taking the factor pair with the largest square number factor, we get √24 = (√4)(√6) = 2√6 ≈ 4.898979.

### Sum-Difference Puzzle:

6 has two factor pairs. One of those factor pairs adds up to 5, and the another one subtracts to 5. Those factor pairs are what you need to solve the first puzzle.

24 has four factor pairs. One of those factor pairs adds up to 10, and another one subtracts to 10. If you know what those factor pairs are, then you can solve the second puzzle.

The second puzzle is really just the first puzzle in disguise. Why would I say that?

### More About the Number 24:

There are 4 counting numbers less than √24. Did you notice that all 4 of those numbers less than √24 are in 24’s factor pairs? OEIS.org informs us that 24 is the LARGEST number that can make that claim.

Another fact: 4! = 1 × 2 × 3 × 4 = 24.

But that’s not all! 24 is a repdigit palindrome in 4 bases!

• 44 BASE 5 because 4(5) + 4(1) = 24.
• 33 BASE 7 because 3(7) + 3(1) = 24.
• 22 BASE 11 because 2(11) + 2(1) = 24.
• 11 BASE 23 because 1(23) + 1(1) = 24.

24 is also the sum of consecutive prime numbers: 11 + 13 = 24.

When 24 is a clue in the FIND THE FACTORS 1-10 Puzzles, the factors will be either 3 × 8 or 4 × 6. When 24 is a clue in the FIND THE FACTORS 1 -12 puzzles, the factors could be 2 × 12, 3 × 8, or 4 × 6. In each case, only one set of factors will be used for each clue in any particular puzzle.