Puzzles

447 and Level 6

/ ivasallay / 4 Comments

447 is the hypotenuse for one Pythagorean triple: 153-420-447. Each of the numbers in the triple are divisible by 3. If you divide each of them by 3, you will discover the Primitive triple that this Pythagorean triple is based on and find one of 447’s factor pairs.

447 Puzzle

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

  • 447 is a composite number.
  • Prime factorization: 447 = 3 x 149
  • 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 447 has exactly 4 factors.
  • Factors of 447: 1, 3, 149, 447
  • Factor pairs: 447 = 1 x 447 or 3 x 149
  • 447 has no square factors that allow its square root to be simplified. √447 ≈ 21.1424

447 Logic

Eight Ways to Make 446 with Three Squares

/ ivasallay / Leave a comment

When I read that 446 is the smallest number that can be expressed as the sum of three squares eight different ways, I just had to find those eight ways for myself. I found them, and it was fun. I made a graphic showing the eight ways:

8 ways to make 446 with 3 squares

446 Puzzle

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

  • 446 is a composite number.
  • Prime factorization: 446 = 2 x 223
  • 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 446 has exactly 4 factors.
  • Factors of 446: 1, 2, 223, 446
  • Factor pairs: 446 = 1 x 446 or 2 x 223
  • 446 has no square factors that allow its square root to be simplified. √446 ≈ 21.1187

446 Logic

 

 

445 and Level 4

/ ivasallay / 11 Comments

Supposedly there is something interesting about every number. What is interesting about the number 445? When you write it in base 9, the digits are reversed. How do you convert 445 from base 10 to base 9?

Solution: The powers of 9 less than 445 in descending order are 81, 9, and 1. We first divide 445 by 81, next we divide the remainder by 9, and lastly, we divide that remainder by 1 as illustrated below.

445 base 9

Thus 445 (base 10) = 544 (base 9).

445 Puzzle

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

  • 445 is a composite number.
  • Prime factorization: 445 = 5 x 89
  • 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 445 has exactly 4 factors.
  • Factors of 445: 1, 5, 89, 445
  • Factor pairs: 445 = 1 x 445 or 5 x 89
  • 445 has no square factors that allow its square root to be simplified. √445 ≈ 21.0950

445 is the hypotenuse of four Pythagorean triples:

  • [267-356-445] which is [3-4-5] times 89
  • [195-400-445] which is [39-80-89] times 5
  • Primitive [84-437-445] (Note: 84 = 2(2 x 21) and 437 = 21^2 – 2^2, while 445 = 21^2 + 2^2)
  • and Primitive [203-396-445] (Note: 396 = 2(18 x 11) and 203 = 18^2 – 11^2, while 445 = 18^2 + 11^2)

445 Logic

 

444 and Level 3

/ ivasallay / 2 Comments

What are the factors of 444? I’m not going to tell you to add 4 + 4 + 4 to see if it is divisible by 3 or 9, because it so obvious, there are three 4’s and three of anything is divisible by 3. The only way three of something could be divisible by 9 is if it were 333 or 666 or 999 so 444 is not divisible by 9.

But try adding the digits for a completely different reason: 4 + 4 + 4 = 12. Guess what, 444 can be evenly divided by 12. Many numbers can be divided evenly by the sum of their digits. In recreational mathematics, numbers with this property are called Harshad numbers. (Saying something is recreational mathematics is NOT an April Fool’s Day joke. Many people actually enjoy mathematics, including me.)

This also is no April Fool’s Day joke: I had a grandson born today at 1:47 pm; He weighed 8 lbs 14 oz.

This puzzle was made for your recreational enjoyment:

444 Puzzle

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

  • 444 is a composite number.
  • Prime factorization: 444 = 2 x 2 x 3 x 37, which can be written 444 = (2^2) x 3 x 37
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 444 has exactly 12 factors.
  • Factors of 444: 1, 2, 3, 4, 6, 12, 37, 74, 111, 148, 222, 444
  • Factor pairs: 444 = 1 x 444, 2 x 222, 3 x 148, 4 x 111, 6 x 74, or 12 x 37
  • Taking the factor pair with the largest square number factor, we get √444 = (√4)(√111) = 2√111 ≈ 21.0713

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.

444 Factors

443 and Level 2

/ ivasallay / Leave a comment

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.

443 Puzzle

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.

443 Factors

442 and Level 1

/ ivasallay / 3 Comments

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

442 sum of squares

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.

442 Puzzle

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

442 Factors

 

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