443 and Level 2

Most people can tell just by looking that 443 cannot be evenly divided by 2, 3, or 5. Could it possibly be a prime number? Scroll down past the puzzle to know for sure.

Print the puzzles or type the factors on this excel file: 10 Factors 2015-03-30

• 443 is a prime number.
• Prime factorization: 443 is prime.
• The exponent of prime number 443 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 443 has exactly 2 factors.
• Factors of 443: 1, 443
• Factor pairs: 443 = 1 x 443
• 443 has no square factors that allow its square root to be simplified. √443 ≈ 21.0476

How do we know that 443 is a prime number? If 443 were not a prime number, then it would be divisible by at least one prime number less than or equal to √443 ≈ 21.0476. Since 443 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 443 is a prime number.

442 and Level 1

442 is the sum of two square numbers two different ways:

Even number 442 is also the sum of consecutive primes. Will the number of consecutive primes be even or odd? If you figure out what those prime numbers are, look in the comments to see if you were right.

Print the puzzles or type the factors on this excel file: 10 Factors 2015-03-30

Here is the factoring information for 442:

• 442 is a composite number.
• Prime factorization: 442 = 2 x 13 x 17
• 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 442 has exactly 8 factors.
• Factors of 442: 1, 2, 13, 17, 26, 34, 221, 442
• Factor pairs: 442 = 1 x 442, 2 x 221, 13 x 34, or 17 x 26
• 442 has no square factors that allow its square root to be simplified. √442 ≈ 21.0238

442 is the hypotenuse of four Pythagorean triples:

• 170-408-442 which is [5-12-13] times 34
• 208-390-442 which is [8-15-17] times 26
• 280-342-442 which is [140-171-221] times 2
• 42-440-442 which is [21-220-221] times 2

441 Consecutive Numbers

The first 6 triangular numbers are

• 1 = 1
• 1 + 2 = 3
• 1 + 2 + 3 = 6
• 1 + 2 + 3 + 4 = 10
• 1 + 2 + 3 + 4 + 5 = 15
• 1 + 2 + 3 + 4 + 5 + 6 = 21

For millennia mathematicians have thought triangular numbers were quite interesting. 21 is a triangular number and 21 x 21 = 441 which makes 441 interesting, too. But wait, there’s something else that is very interesting about triangular numbers:

It is amazing that when we begin with 1 cube, the sum of n consecutive cubes equals the nth triangular number squared every time!

What  follows is less amazing, but very practical. We can add consecutive numbers several ways to get 441. To find those ways we need to know the factors of 441.

• 441 is a composite number.
• Prime factorization: 441 = 3 x 3 x 7 x 7, which can be written 441 = (3^2) x (7^2)
• The exponents in the prime factorization are 2 and 2. Adding one to each and multiplying we get (2 + 1)(2 + 1) = 3 x 3 = 9. Therefore 441 has exactly 9 factors.
• Factors of 441: 1, 3, 7, 9, 21, 49, 63, 147, 441
• Factor pairs: 441 = 1 x 441, 3 x 147, 7 x 63, 9 x 49, or 21 x 21
• 441 is a perfect square. √441 = 21

Because all of the factors for 441 are odd numbers, it is so easy to find consecutive numbers whose sum equal 441. Check out all of these:

If we allowed negative numbers in the list of consecutive numbers we could also see that 441 equals the sum of 147 consecutive numbers that are centered around the number 3, and 49 consecutive numbers that are centered around the number 7, and 63 consecutive numbers that are centered around the number 9. All of those sums would be quite long.

Here are some more reasonable-length sums using only consecutive ODD numbers.

That last sum reminds us that we always get n squared when we begin with one and add n consecutive odd numbers together.

440 Can You Appreciate a Good Paradox?

Brainden.com gives several examples of paradoxes as well as this definition: “A paradox is a statement that contradicts itself or a situation which seems to defy logic.”

Alan Parr suggested I write no more than the fact that 440 is the sum of consecutive prime numbers. It may seem to defy logic that he would want to try to figure out what those consecutive primes are without any hints, but he proved yesterday he is up for the challenge. Check the comments to see if he (or somebody else) can figure out what those prime numbers might be. I am waiting to see if he can again defy logic and figure them out. In the meantime, the factors of 440 will be listed at the end of this post.

I remember watching the episode of Star Trek in which several androids malfunctioned because they couldn’t handle paradoxes, but most humans seem to be able to handle paradoxes just fine. I’ve written one myself that I hope you will enjoy:

I overheard a pair of docs in the middle of a heated argument. One of them said, “As I have stated hundreds of times before, no matter what I say, YOU ALWAYS contradict me!” The other doc shook his head saying, “No, no. That’s not correct at all. YOU’RE the one who’s always contradicting ME!” I hate to hear arguing, so I left, but I have to wonder if they ever worked it out or are they still arguing about contradicting each other even today?

You might not expect to find paradoxes in the The New Testament, but crosswalkblogdotcom describes several of them beautifully in I-will-take-paradox-for-200-alec.

Shakespeare and Wordsworth used them as well. See examples in Is-this-phrase-grammatically-and-semantically-correct?

When I first joined twitter, this is part of the first conversation I had:

Steve Morris has written about paradoxes on several occasions including his very popular post titled Failure-to-fail. If we fail to fail, we become the biggest failure.

It is a paradox to have to be willing to fail in order to really succeed. As Michael Jordan stated in a Nike commercial: “I’ve missed more than 9000 shots in my career. I’ve lost almost 300 games. 26 times, I’ve been trusted to take the game winning shot and missed. I’ve failed over and over and over again in my life. And that is why I succeed.”

We will face setbacks, trials of faith, and other paradoxes in our lives, but we must not give up. All of them are opportunities for us to learn and grow. We actually ought to be grateful for them, because as we face them, we become stronger and more successful. Zeno’s paradox would have us believe that we can never reach our goals. That is simply not true. We can reach them if we keep them in our sights and keep going toward them relentlessly.

Here are the factors of 440:

• 440 is a composite number.
• Prime factorization: 440 = 2 x 2 x 2 x 5 x 11, which can be written 440 = (2^3) x 5 x 11
• The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 440 has exactly 16 factors.
• Factors of 440: 1, 2, 4, 5, 8, 10, 11, 20, 22, 40, 44, 55, 88, 110, 220, 440
• Factor pairs: 440 = 1 x 440, 2 x 220, 4 x 110, 5 x 88, 8 x 55, 10 x 44, 11 x 40, or 20 x 22
• Taking the factor pair with the largest square number factor, we get √440 = (√4)(√110) = 2√110 ≈ 20.9762

Since 440 = 20 × 22, we know that 441 = 21 × 21.

439 and Level 6

439 is a prime number, and it is also the sum of consecutive prime numbers in TWO different ways. Can you find either or both of those ways? You can write your answer or give or ask for hints in the comments.

In case someone is not sure what consecutive prime numbers are, here is an example: 17, 19, 23, 29, 31, and 37 are six consecutive prime numbers because they are ALL the prime numbers from 17 to 37 and they are listed in order. If we took their sum, we would get 156.

Print the puzzles or type the factors on this excel file: 12 Factors 2015-03-23

• 439 is a prime number.
• Prime factorization: 439 is prime.
• The exponent of prime number 439 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 439 has exactly 2 factors.
• Factors of 439: 1, 439
• Factor pairs: 439 = 1 x 439
• 439 has no square factors that allow its square root to be simplified. √439 ≈ 20.9523

How do we know that 439 is a prime number? If 439 were not a prime number, then it would be divisible by at least one prime number less than or equal to √439 ≈ 20.9523. Since 439 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 439 is a prime number.

438 and Level 5

The sum of the digits of 438 is 4 + 3 + 8 = 15. The sum of the digits of its prime factorization (2 x 3 x 73) is 2 + 3 + 7 + 3 = 15. Since the two sums are equal, 438 is the 18th Smith number.

Print the puzzles or type the factors on this excel file: 12 Factors 2015-03-23

• 438 is a composite number.
• Prime factorization: 438 = 2 x 3 x 73
• 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 438 has exactly 8 factors.
• Factors of 438: 1, 2, 3, 6, 73, 146, 219, 438
• Factor pairs: 438 = 1 x 438, 2 x 219, 3 x 146, or 6 x 73
• 438 has no square factors that allow its square root to be simplified. √438 ≈ 20.9284

437 and Level 4

In a month my husband and I will celebrate our 38th wedding anniversary. David Mitchell has given us an early and lovely present, wedding anniversary tessellations. Tessellations are quite fascinating and the ones he makes are very intricate and well worth a look.

The Factors of 437 are listed below the puzzle.

Print the puzzles or type the factors on this excel file: 12 Factors 2015-03-23

• 437 is a composite number.
• Prime factorization: 437 = 19 x 23
• 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 437 has exactly 4 factors.
• Factors of 437: 1, 19, 23, 437
• Factor pairs: 437 = 1 x 437 or 19 x 23
• 437 has no square factors that allow its square root to be simplified. √437 ≈ 20.9045

436 Pythagorean Triples and Level 3

Probably the most famous Pythagorean triple is 3-4-5. It doesn’t matter if the 3-4-5 refers to millimeters, centimeters, inches, meters, or even miles, this relationship will always hold: 3^2 + 4^2 = 5^2.

Likewise, the relationship will still hold if we multiply each number 3-4-5 by the same number. Let’s multiply each of them by 109 to get 327-436-545. Go ahead and test it: 327^2 + 436^2 = 545^2.

The following graphic shows a Pythagorean triple with numbers much larger than the familiar 3-4-5. Still everything is relative: 60-91-109 in millimeters is much smaller than 3-4-5 in miles.

You may have noticed that 436 appeared in a couple of those triples I listed above. That’s because 436 = 4 x 109. Some of those side lengths may have seemed large, but try imagining this Pythagorean triple: 436-23760-23764 or this even larger primitive triple: 436-47523-47525. In each of those monstrous cases, the sum of the squares of the first two numbers equals the square of the last number, too!

This puzzle is relatively easy to solve:

Print the puzzles or type the factors on this excel file: 12 Factors 2015-03-23

• 436 is a composite number.
• Prime factorization: 436 = 2 x 2 x 109, which can be written 436 = (2^2) x 109
• 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 436 has exactly 6 factors.
• Factors of 436: 1, 2, 4, 109, 218, 436
• Factor pairs: 436 = 1 x 436, 2 x 218, or 4 x 109
• Taking the factor pair with the largest square number factor, we get √436 = (√4)(√109) = 2√109 ≈ 20.8806

A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

435 is a Hexagonal Number

It isn’t too hard to visualize the first three hexagonal numbers and count 1, 6, or 15 dots. Any more than that boggles my mind so I made a less mind-boggling chart with the first fifteen hexagonal numbers. In the picture on the chart you can see the first dot that makes the first hexagonal number and several other hexagonal numbers embedded in the the seventh hexagonal number that has 91 dots. Counting all those dots can be quite tedious!

The nth hexagonal number = n(2n – 1). That means 435 (the 15th hexagonal number) can be written 435 = 15 (2 x 15 – 1) = 15 x 29.

435 is also the 29th triangular number because 435 = 29 (29 + 1)/2 = 29 x 15. (All hexagonal numbers are also triangular numbers.)  Being the 29th triangular number means 1 + 2 + 3 + 4 + . . . + 26 + 27 + 28 + 29 = 435

The other factors of 435 are listed below the puzzle.

Print the puzzles or type the factors on this excel file: 12 Factors 2015-03-23

• 435 is a composite number.
• Prime factorization: 435 = 3 x 5 x 29
• 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 435 has exactly 8 factors.
• Factors of 435: 1, 3, 5, 15, 29, 87, 145, 435
• Factor pairs: 435 = 1 x 435, 3 x 145, 5 x 87, or 15 x 29
• 435 has no square factors that allow its square root to be simplified. √435 ≈ 20.8567

434 Going Backwards Is Important!

The factors of 434 are listed below the puzzle, but here is a quick problem (really a puzzle) for you to figure out:

434 is the sum of six consecutive prime numbers. One of those primes is 71. What are the other five? If you would like, you may type your answer in the comments.

Solving that problem will require you to work backwards.

Many students may think they understand a mathematical concept, but they don’t really understand it unless they can also do it backwards. For example, they may understand, but maybe not remember that 7 x 8 = 56. Yet, these same students are completely baffled if you tell them this rhyme, “Five, six, seven, eight, 56 is 7 times 8.” They aren’t used to seeing addition, subtraction, or multiplication facts written or spoken in that backwards way. Coleen Young made an excellent slide presentation titled good-mathematicians-can-go-backwards. We ALL can go forward in mathematics so much more confidently once we learn to go backwards.

Coleen recommends many resources including the Find The Factors puzzles. Each of these puzzles can be solved and then completely filled out as a multiplication table if you will go backwards first and find the factors of the clues in the grid. Think what two numbers multiplied together give you 49? Give it a try!

Print the puzzles or type the factors on this excel file: 12 Factors 2015-03-23

• 434 is a composite number.
• Prime factorization: 434 = 2 x 7 x 31
• 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 434 has exactly 8 factors.
• Factors of 434: 1, 2, 7, 14, 31, 62, 217, 434
• Factor pairs: 434 = 1 x 434, 2 x 217, 7 x 62, or 14 x 31
• 434 has no square factors that allow its square root to be simplified. √434 ≈ 20.8327