Prime number

355 and Level 5

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Five can evenly divide into numbers when the last digit is 5, so 355 is a composite number. Scroll down past the puzzle to see its factors.

355 Puzzle

Print the puzzles or type the factors on this excel file: 12 Factors 2015-01-12

  • 355 is a composite number.
  • Prime factorization: 355 = 5 x 71
  • 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 355 has exactly 4 factors.
  • Factors of 355: 1, 5, 71, 355
  • Factor pairs: 355 = 1 x 355 or 5 x 71
  • 355 has no square factors that allow its square root to be simplified. √355 ≈ 18.841

355 Logic

354 and Level 4

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354 is even and thus divisible by 2, so 354 is a composite number. It is also made from three consecutive numbers so it is divisible by 3. Scroll down past the puzzle to see its factors.

354 Puzzle

Print the puzzles or type the factors on this excel file: 12 Factors 2015-01-12

  • 354 is a composite number.
  • Prime factorization: 354 = 2 x 3 x 59
  • 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 354 has exactly 8 factors.
  • Factors of 354: 1, 2, 3, 6, 59, 118, 177, 354
  • Factor pairs: 354 = 1 x 354, 2 x 177, 3 x 118, or 6 x 59
  • 354 has no square factors that allow its square root to be simplified. √354 ≈ 18.815

354 Logic

353 and Level 3

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When 2^353 is divided by 353, the remainder is 2, so 353 is VERY LIKELY a prime number. Scroll down past the puzzle to know for sure.

353 Puzzle

Print the puzzles or type the factors on this excel file: 12 Factors 2015-01-12

  • 353 is a prime number.
  • Prime factorization: 353 is prime.
  • The exponent of prime number 353 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 353 has exactly 2 factors.
  • Factors of 353: 1, 353
  • Factor pairs: 353 = 1 x 353
  • 353 has no square factors that allow its square root to be simplified. √353 ≈ 18.788

How do we know that 353 is a prime number? If 353 were not a prime number, then it would be divisible by at least one prime number less than or equal to √353 ≈ 18.788. Since 353 cannot be divided evenly by 2, 3, 5, 7, 11, 13, or 17, we know that 353 is a prime number.

A Logical Approach to FIND THE FACTORS: 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.

353 Factors

Square Roots up to √352 That Can Be Simplified

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  • 352 is even and therefore a composite number.
  • Prime factorization: 352 = 2 x 2 x 2 x 2 x 2 x 11, which can be written 352 = (2^5) x 11
  • The exponents in the prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 x 2 = 12. Therefore 352 has exactly 12 factors.
  • Factors of 352: 1, 2, 4, 8, 11, 16, 22, 32, 44, 88, 176, 352
  • Factor pairs: 352 = 1 x 352, 2 x 176, 4 x 88, 8 x 44, 11 x 32, or 16 x 22
  • Taking the factor pair with the largest square number factor, we get √352 = (√16)(√22) = 4√22 ≈ 18.762

Mathematics is full of interesting patterns. Let’s explore some patterns in reducible square roots.

Here are the first ten numbers that have reducible square roots. Notice that five of the numbers in this list, 1, 4, 9, 16, and 25 are perfect squares:

1-25 sqrt

40% of the numbers up to 5 have reducible square roots. The same thing is true for 40% of the numbers up to 10, 40% of the numbers up to 20, and 40% of the numbers up to 25.

Here are the second ten numbers with reducible square roots. The last number,50, is double the previous last number, 25. Again 40% of the numbers up to 50 have reducible square roots, and only two of these numbers are perfect squares. Notice that the last three numbers under the radical sign in this set are consecutive numbers.

27-50 sqrt

Would you like to make any predictions for the third set of ten numbers with reducible square roots?

If you predicted that this set of numbers would end with 75, you were almost right! 29/75 or 38.67% of the numbers up to 75 have reducible square roots, and 30/76 or 39.47% of the numbers up to 76 do. Both of these values are very close to 40%, but not quite there. Notice this time only one number is a perfect square.

52-76 sqrt

Would you like to make a prediction for what will happen with the fourth set of ten numbers with reducible square roots?

Surprise! We’re back to 40% of the numbers up to 100 have reducible square roots. Those consecutive numbers at the end of the set really helped raise the percentage right at the last minute. This set of numbers has two perfect squares.

80-100 sqrt

What do you think will happen if we look at all the reducible square roots up to 350? Multiples of any perfect square will always have reducible square roots. Since we have more perfect squares, do you think more will be reducible?

Because the square roots of 351 and 352 are also reducible, let’s include them in this chart. Each column has 20 reducible square roots in it, and they are grouped into fives for easier counting. I’ve highlighted sets of three or four consecutive numbers. In all, we now have charts showing the first 140 reducible square roots. This last set has eight perfect squares. Let’s look at the percentages at the end of some of those sets of consecutive numbers: 50/126 or 39.68% of the first 126 numbers have reducible square roots. 96/245 or 39.18% of the first 245 numbers have reducible square roots. Finally, 140/352 or 39.77% of the numbers up to 352 have reducible square roots. That one is so close to 40%!

104-352 sqrt

Sets of 41-60, 61-80, 81-100, 101-120, & 121-140 Reducible Square Roots

No matter how big a table we make, the percentage of reducible square roots will be very close to 40%. It will not get significantly higher because, believe it or not, most numbers are either prime numbers or the product of two or more DIFFERENT primes and thus have square roots that are NOT reducible.

351 is a Triangular Number

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351 is the 26th triangular number. The only triangular numbers that are NOT composite numbers are 1 and 3. All other triangular numbers are composite numbers because they can be written as the product of two consecutive numbers divided by 2. For example, 351 can be written like this:

  • 1 + 2 + 3 + . . . + 24 + 25 + 26 = (26 x 27)/2 = 351

Also, 351 is divisible by 3 because 1, 3, 5 are three consecutive odd numbers. Since 3 is the middle number in that list, 351 is also divisible by 9. Scroll down past the puzzle to see the rest of 351’s factors.

351 Puzzle

Print the puzzles or type the factors on this excel file: 12 Factors 2015-01-12

  • 351 is a composite number.
  • Prime factorization: 351 = 3 x 3 x 3 x 13, which can be written (3^3) x 13
  • 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 351 has exactly 8 factors.
  • Factors of 351: 1, 3, 9, 13, 27, 39, 117, 351
  • Factor pairs: 351 = 1 x 351, 3 x 117, 9 x 39, or 13 x 27
  • Taking the factor pair with the largest square number factor, we get √351 = (√9)(√39) = 3√39 ≈ 18.735

351 is in this cool pattern:

351 Factors

350 and Level 1

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350 is even and thus divisible by 2, so 350 is a composite number. Scroll down past the puzzle to see its factors.

In pose-a-puzzle Ed Southall (solvemymaths) showed examples and described several great mathematical puzzles that are available on the internet. He wrote, “Not only do they excite students in a way that perhaps no textbook page can, they also make you think. I don’t think any good puzzles can be done just by applying a mindless algorithm to a familiar looking structure.”

It will take a while, but check out all the different types of puzzles listed. Here’s something he wrote specifically about the Find the Factors puzzles: “They are the best way I’ve found to get students of *any* ability to practice multiplication tables.”

350 Puzzle

Print the puzzles or type the factors on this excel file: 12 Factors 2015-01-12

  • 350 is a composite number.
  • Prime factorization: 350 = 2 x 5 x 5 x 7, which can be written 350 = 2·5²·7
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 x 3 x 2 = 12. Therefore 350 has exactly 12 factors.
  • Factors of 350: 1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175, 350
  • Factor pairs: 350 = 1 x 350, 2 x 175, 5 x 70, 7 x 50, 10 x 35, or 14 x 25
  • Taking the factor pair with the largest square number factor, we get √350 = (√14)(√25) = 5√14 ≈ 18.708

349 and Level 6

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When 2^349 is divided by 349, the remainder is 2, so 349 is VERY LIKELY a prime number. Scroll down past the puzzle to know for sure.

349 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-01-05

  • 349 is a prime number.
  • Prime factorization: 349 is prime.
  • The exponent of prime number 349 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 349 has exactly 2 factors.
  • Factors of 349: 1, 349
  • Factor pairs: 349 = 1 x 349
  • 349 has no square factors that allow its square root to be simplified. √349 ≈ 18.682

How do we know that 349 is a prime number? If 349 were not a prime number, then it would be divisible by at least one prime number less than or equal to √349 ≈ 18.682. Since 349 cannot be divided evenly by 2, 3, 5, 7, 11, 13, or 17, we know that 349 is a prime number.

Prime Number Trivia: There are sixteen prime numbers between 300 and 400. Of those, 349 is the only one that doesn’t produce another prime number when 300 is subtracted from it. 349 – 300 = 49, a composite number. This piece of trivia might be helpful if you wanted to memorize all the prime numbers less than 1000.

349 Logic

348 and Level 5

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Quick Prime/Composite Number Test: 348 is even and thus divisible by 2, so 348 is a composite number.

348 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-01-05

  • 348 is a composite number.
  • Prime factorization: 348 = 2 x 2 x 3 x 29, which can be written 348 = (2^2) x 3 x 29
  • 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 348 has exactly 12 factors.
  • Factors of 348: 1, 2, 3, 4, 6, 12, 29, 58, 87, 116, 174, 348
  • Factor pairs: 348 = 1 x 348, 2 x 174, 3 x 116, 4 x 87, 6 x 58, or 12 x 29
  • Taking the factor pair with the largest square number factor, we get √348 = (√4)(√87) = 2√87 ≈ 18.655

348 Logic

Prime Number Tests for 347

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  • 347 is a prime number.
  • Prime factorization: 347 is prime.
  • The exponent of prime number 347 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 347 has exactly 2 factors.
  • Factors of 347: 1, 347
  • Factor pairs: 347 = 1 x 347
  • 347 has no square factors that allow its square root to be simplified. √347 ≈ 18.628

How do we know that 347 is a prime number? If 347 were not a prime number, then it would be divisible by at least one prime number less than or equal to √347 ≈ 18.628. Since 347 cannot be divided evenly by 2, 3, 5, 7, 11, 13, or 17, we know that 347 is a prime number.

Is there a quicker way to tell that 347 is a prime number? Not exactly…There is a quick test that will tell you if 347 is VERY LIKELY prime.

Quick Prime or Composite Number Test: You can use as few as TEN calculator keystrokes to test if 347 is VERY LIKELY prime: What is the remainder when 2³⁴⁷ is divided by 347? To find out type “(2)(xʸ)(3)(4)(7)(Mod)(3)(4)(7)” followed by the equal sign into your computer’s calculator. This is how the calculator should look before you hit the equal sign:

mod 347

2^347 mod (347) = 2, the same as the base we typed in. That means 347 is VERY LIKELY prime. This tests works for all prime numbers, but sometimes it gives a false positive for a relatively few (but infinite number) of composite numbers. (341 is the smallest of these numbers, and I wrote about it here.)

2 isn’t the only prime number that can be used as the base in this quick test. You can verify the following also on your computer’s calculator:

  • 3^347 mod (347) = 3
  • 5^347 mod (347) = 5
  • 7^347 mod (347) = 7
  • 11^347 mod (347) = 11. We can practically go on forever using prime number after prime number….until
  • 337^347 mod (347) = 337, the largest prime number less than 347.

As long as we use a prime less than 347 as the base and 347 as the exponent, we will always get that prime as the remainder. Still even though 347 passes ALL those prime number tests, we can only conclude that 347 is VERY LIKELY prime. Doing all those calculations is more work than simply dividing 347 by all the prime numbers less than or equal to its square root and getting a remainder every single time.

Tomcircle, a fellow blogger, shared a video of a new-prime-number-test that works for prime numbers and NEVER gives a false positive.

While this test works every time in theory, it can be quite a nightmare in practice. It involves putting the possible prime number into a particular expression, expanding the expression, calculating each coefficient, and verifying that all those coefficients are divisible by this possible prime number. In the case of the relatively small number 347, there would be 173 different coefficients. Each of those coefficients are the numbers in the 347th row of Pascal’s triangle, and too many of them are usually expressed in scientific notation. Dividing each of them by 347 to verify there is no remainder is far more work than most people would want to do. In fact, dividing 347 by all the integers less than 347 would actually be less work!

The Sieve of Eratosthenes was how the ancient Greeks found prime numbers. It takes a little longer than dividing the possible prime number by all of the primes less than or equal to its square root, but it finds many primes at the same time. Solvemymath shares some interesting facts about it.

Prime numbers are intriguing, and 347 is one of them!

346 and Level 4

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Quick Prime/Composite Number Test: 346 is even and thus divisible by 2, so 346 is a composite number.

346 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-01-05

  • 346 is a composite number.
  • Prime factorization: 346 = 2 x 173
  • 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 346 has exactly 4 factors.
  • Factors of 346: 1, 2, 173, 346
  • Factor pairs: 346 = 1 x 346 or 2 x 173
  • 346 has no square factors that allow its square root to be simplified. √346 ≈ 18.601

346 Logic