number puzzle

439 and Level 6

/ ivasallay / 6 Comments

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.

439 Puzzle

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

/ ivasallay / Leave a comment

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.

438 Puzzle

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

438 Logic

437 and Level 4

/ ivasallay / 6 Comments

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.

437 Difference of 2 squares

437 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

437 Logic

436 Pythagorean Triples and Level 3

/ ivasallay / 1 Comment

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.

60-91-109

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:

436 Puzzle

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.

436 Factors

 

 

435 is a Hexagonal Number

/ ivasallay / 1 Comment

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!

hexagonal numbers and differences

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.

435 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

 435 Factors

434 Going Backwards Is Important!

/ ivasallay / 12 Comments

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!

 434 Puzzle

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

434 Factors

How to Simplify √432

/ ivasallay / 2 Comments

432 factor tree

Many people use factor trees and prime factorizations to figure out how to simplify square roots. I don’t. I don’t see the point in breaking something completely apart just to put it back together again, especially when it ISN’T necessary to break it completely apart.

So instead of using a factor tree, I’ve modified the cake method, and I use it to simplify square roots.

For example, to find the square root of 500, I would never use its prime factorization: 500 = 2 x 2 x 5 x 5 x 5. Instead, I would first divide 500 by perfect square 100 to get 5. Then I would take the square root of both 100 and 5 to get 10√5.

Only 1% of numbers are divisible by 100, but for the ones that are, I always start by dividing by 100.

Now get this: Roughly 82.5% of all numbers that have simplifiable square roots can be evenly divided by perfect squares four and/or by nine. It is so easy to tell if a number can be evenly divided by either of those numbers. That is why I always start with one hundred, then four, then nine.

Let’s look at 432: Since the number formed from the last two digits, 32, is divisible by 4, I know that 432 is also divisible by four, so I will go ahead and do the division.

432 divided by 4

Now I look at 108. The last two digits, 08, can also be evenly divided by 4, so I do that division as well:

108 divided by 4

108 divided by 4 gives us 27 which is not divisible by 4. However since 2 + 7 = 9, I know that 27 is divisible by 9, so I do that division next:

27 divided by 9

27 divided by 9 is 3. Since the only square number that will divide evenly into 3 is 1, I’m done with the division process. Now I take the square roots of all the numbers on the outside of the cake and multiply them together: √(4 x 4) x (√9) x (√3) = 4 x 3 x (√3) = 12√3.

I will give other examples of this method in future posts. Here’s today’s puzzle:

432 Puzzle

Print the puzzles or type the factors on this excel file:10 Factors 2015-03-16

  • 432 is a composite number.
  • Prime factorization: 432 = 2 x 2 x 2 x 2 x 3 x 3 x 3, which can be written 432 = (2^4) x (3^3)
  • The exponents in the prime factorization are 4 and 3. Adding one to each and multiplying we get (4 + 1)(3 + 1) = 5 x 4 = 20. Therefore 432 has exactly 20 factors.
  • Factors of 432: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 108, 144, 216, 432
  • Factor pairs: 432 = 1 x 432, 2 x 216, 3 x 144, 4 x 108, 6 x 72, 8 x 54, 9 x 48, 12 x 36, 16 x 27, or 18 x 24
  • Taking the factor pair with the largest square number factor, we get √432 = (√144)(√3) = 12√3 ≈ 20.7846

432 Logic

431 and Level 5

/ ivasallay / 4 Comments

431 is the sum of seven consecutive prime numbers beginning with prime number 47. What other prime numbers are in that sum? If you would like, you can type your answer in the comments.

2^431 (mod 431) = 2 so 431 is VERY likely a prime number. Scroll down past the puzzle to know for sure.

431 Puzzle

Print the puzzles or type the factors on this excel file:10 Factors 2015-03-16

  • 431 is a prime number.
  • Prime factorization: 431 is prime.
  • The exponent of prime number 431 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 431 has exactly 2 factors.
  • Factors of 431: 1, 431
  • Factor pairs: 431 = 1 x 431
  • 431 has no square factors that allow its square root to be simplified. √431 ≈ 20.7605

How do we know that 431 is a prime number? If 431 were not a prime number, then it would be divisible by at least one prime number less than or equal to √431 ≈ 20.7605. Since 431 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 431 is a prime number.

431 Logic

430 and Level 4

/ ivasallay / 4 Comments

The factors of 430 are listed after the puzzle.

What interesting fact could I say about the number 430? I did a little research.

Biblewheel.com quotes a verse from Galatians 3:17 in the Bible: “And this I say, that the covenant, that was confirmed before of God in Christ, the law, which was four hundred and thirty years after, cannot disannul, that it should make the promise of none effect.” Biblewheel states that 430 means the law, just as 666 means the beast.

OEIS.org says that 430 “is the number of necklaces possible with 6 beads, each being one of 4 colors.” If I understood what that means in English, I’m sure I could construct it mathematically. I know it can’t mean 6 Choose 4. If you understand what it means, please comment.

430 Puzzle

Print the puzzles or type the factors on this excel file:10 Factors 2015-03-16

  • 430 is a composite number.
  • Prime factorization: 430 = 2 x 5 x 43
  • 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 430 has exactly 8 factors.
  • Factors of 430: 1, 2, 5, 10, 43, 86, 215, 430
  • Factor pairs: 430 = 1 x 430, 2 x 215, 5 x 86, or 10 x 43
  • 430 has no square factors that allow its square root to be simplified. √430 ≈ 20.7364

430 Logic

Level 3 and Divisibility Tricks Applied to 429

/ ivasallay / 11 Comments

Today my blog has had 42 referral views from twitter, mostly thanks to other people’s tweets. Thank you for the tweets and retweets. Also, welcome everyone who visits my blog because of them or for any other reason!

429 Puzzle

Print the puzzles or type the factors on this excel file:10 Factors 2015-03-16

Let’s try some divisibility tricks on the number 429:

4 + 2 is 6, a multiple of three, so 429 is a divisible by three. Why didn’t I include 9 in the sum? Because I already know that 9 is a multiple of three, so it is redundant to include it!

Now try another divisibility trick on 429: I’ve put the odd numbered digits in bold and the even numbered digits in regular type. If we sum the odd numbered digits and subtract the even numbered digit, we get 4 + 9 – 2 = 11, a multiple of 11. That means that 429 is also divisible by 11!

  • 429 is a composite number.
  • Prime factorization: 429 = 3 x 11 x 13
  • 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 429 has exactly 8 factors.
  • Factors of 429: 1, 3, 11, 13, 33, 39, 143, 429
  • Factor pairs: 429 = 1 x 429, 3 x 143, 11 x 39, or 13 x 33
  • 429 has no square factors that allow its square root to be simplified. √429 ≈ 20.7123

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.

429 Factors