number puzzle

Square Roots Up to √513 That Can Be Simplified

/ ivasallay / 2 Comments

5 + 1 + 3 = 9 so 513 can be evenly divided by 9, and thus its square root can be simplified.

513 is the 200th counting number whose square root can be reduced. 200/513 ≈ .38986, which means, so far, 38.97% of the counting numbers have reducible square roots.

Here are the first 100 reducible square roots followed by the second hundred:

1st 100 reducible square roots

I highlighted the ones that are part of three or more consecutive reducible square roots.

2nd 100 reducible square roots

  • 513 is a composite number.
  • Prime factorization: 513 = 3 x 3 x 3 x 19, which can be written 513 = (3^3) x 19
  • 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 513 has exactly 8 factors.
  • Factors of 513: 1, 3, 9, 19, 27, 57, 171, 513
  • Factor pairs: 513 = 1 x 513, 3 x 171, 9 x 57, or 19 x 27
  • Taking the factor pair with the largest square number factor, we get √513 = (√9)(√57) = 3√57 ≈ 22.6495033

512 Ants and Level 4

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The 512 Ants on Sullivan Street

 

The 512 Ants on Sullivan Street is an entertaining book for children that also includes learning activities by Marilyn Burns. The book teaches the important concept of doubling in a very natural way, but it can teach even more than that.  Here is a place value lesson for third grade students that uses the same book. There are relationships between adding, subtracting, doubling, multiplying, dividing, powers, and place value. The book and the place value lesson will make some of those relationships clear.

512 is a perfect cube because 512 = 8^3. It is also a cube of a cube because 8 = 2^3. Thus 512 = (2^3)^3 = 2^9. All of the factors of 512 are powers of 2.

512 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-01

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  • 512 is a composite number.
  • Prime factorization: 512 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2, which can be written 512 = (2^9)
  • The exponent in the prime factorization is 9. Adding one we get (9 + 1) = 10. Therefore 512 has exactly 10 factors.
  • Factors of 512: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512
  • Factor pairs: 512 = 1 x 512, 2 x 256, 4 x 128, 8 x 64, or 16 x 32
  • Taking the factor pair with the largest square number factor, we get √512 = (√256)(√2) = 16√2 ≈ 22.627417

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512 Logic

511 Collaborators

/ ivasallay / 4 Comments

Hungarian Paul Erdős was one of the most famous mathematicians of the twentieth century. Mathematics was a social activity for him, and fellow mathematicians absolutely loved associating with him and collaborating with him. Although he wrote some mathematical papers by himself, most of his approximately 1,525 articles were collaborations with other mathematicians. There are 511 people with an Erdős number of one, which means that 511 people had the privilege of writing at least one paper with him. 511 represents an astounding number of people.

Since I find both Hungary and mathematics quite fascinating, I especially loved reading all about Hungarian mathematician Paul Erdős in the book The Man Who Loved Only Numbers. I wish I could have heard him speak while he was still alive.

511 is also the hypotenuse of the Pythagorean triple 336-385-511. Can you find the greatest common factor of those three numbers?

Can you fill in the factors of the following puzzle?

511 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-01

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  • 511 is a composite number.
  • Prime factorization: 511 = 7 x 73
  • 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 511 has exactly 4 factors.
  • Factors of 511: 1, 7, 73, 511
  • Factor pairs: 511 = 1 x 511 or 7 x 73
  • 511 has no square factors that allow its square root to be simplified. √511 ≈ 22.60530911

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510 and Level 2

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510 is the sum of consecutive prime numbers three different ways.

  1. 510 = 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79
  2. 510 = 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71
  3. 510 = 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67

510 is the hypotenuse of four Pythagorean triples, none of which are primitive:

  1. 78-504-510
  2. 216-462-510
  3. 240-450-510
  4. 306-408-510

Two of those triples have the same greatest common factor. Can you identify which ones they are?

510 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-01

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  • 510 is a composite number.
  • Prime factorization: 510 = 2 x 3 x 5 x 17
  • 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 510 has exactly 16 factors.
  • Factors of 510: 1, 2, 3, 5, 6, 10, 15, 17, 30, 34, 51, 85, 102, 170, 255, 510
  • Factor pairs: 510 = 1 x 510, 2 x 255, 3 x 170, 5 x 102, 6 x 85, 10 x 51, 15 x 34, or 17 x 30
  • 510 has no square factors that allow its square root to be simplified. √510 ≈ 22.58317958

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510 Factors

509 and Level 1

/ ivasallay / Leave a comment

509 = 22^2 + 5^2.

509 is the hypotenuse of the primitive Pythagorean triple, 220-459-509.

509 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-01

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  • 509 is a prime number.
  • Prime factorization: 509 is prime.
  • The exponent of prime number 509 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 509 has exactly 2 factors.
  • Factors of 509: 1, 509
  • Factor pairs: 509 = 1 x 509
  • 509 has no square factors that allow its square root to be simplified. √509 ≈ 22.561028

How do we know that 509 is a prime number? If 509 were not a prime number, then it would be divisible by at least one prime number less than or equal to √509 ≈ 22.6. Since 509 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 509 is a prime number.

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509 Factors

507 and Level 6

/ ivasallay / Leave a comment

507 cannot be evenly divided by 4 or 9, but to simplify its square root, I would still make a little cake:

507 cake

√507 = (√169)(√3) = 13√3

If I didn’t recognize that 169 is a perfect square, I would apply some prime number divisibility tricks in numerical order on 169:

  • 2: 169 is not even so it’s not divisible by 2.
  • 3: 1 + 6 + 9 = 16 which is not a multiple of 3 so 169 is not divisible by 3.
  • 5: The last digit is not 0 or 5, so 169 cannot be evenly divided by 5.
  • 7: The difference between 16 and 9 x 2 is 2 which is not a divisible by 7, so 169 is not divisible by 7.
  • 11: 1 – 6 + 9 = 4 which is not a multiple of 11, so 169 is not divisible by 11.
  • 13: To check if 169 is divisible by 13, let’s make a second layer to our cake:

507 2-layer cake

Taking the square root of everything on the outside of the cake and multiplying it all together we get, √507 = (√3)(√13)(√13) = 13√3.

Here’s today’s puzzle:

507 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-25

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  • 507 is a composite number.
  • Prime factorization: 507 = 3 x 13 x 13, which can be written 507 = 3 x (13^2)
  • 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 507 has exactly 6 factors.
  • Factors of 503: 1, 3, 13, 39, 169, 507
  • Factor pairs: 507 = 1 x 507, 3 x 169, or 13 x 39
  • Taking the factor pair with the largest square number factor, we get √507 = (√169)(√3) = 13√3 ≈ 22.51666

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507 Logic

506 and Level 5

/ ivasallay / Leave a comment

506 is divisible by 11 because 5 + 6 – 0 = 11, and 11 obviously is divisible by 11.

506 is the 11th square pyramidal number because it is the sum of the first eleven square numbers.

Thus 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 + 121 = 506.

That was predictable because 506 = (11 x 12 x 23)/6 and 12 = 11 + 1 and 23 = 2(11) + 1.

Since 506 = 22 x 23, it is the sum of the first 22 even numbers which also happens to be exactly two times the 22nd triangular number, 253.

Now here’s a Level 5 puzzle for you to try:

506 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-25

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  • 506 is a composite number.
  • Prime factorization: 506 = 2 x 11 x 23
  • 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 506 has exactly 8 factors.
  • Factors of 506: 1, 2, 11, 22, 23, 46, 253, 506
  • Factor pairs: 506 = 1 x 506, 2 x 253, 11 x 46, or 22 x 33
  • 506 has no square factors that allow its square root to be simplified. √506 ≈ 22.49444

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506 Logic

505 and Level 4

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I learned the coolest visual fact about the number 505 from OEIS.org:

505 = 10C5 + 10C0 + 10C5.

505 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-25

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  • 505 is a composite number.
  • Prime factorization: 505 = 5 x 101
  • 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 505 has exactly 4 factors.
  • Factors of 505: 1, 5, 101, 505
  • Factor pairs: 505 = 1 x 505 or 5 x 101
  • 505 has no square factors that allow its square root to be simplified. √505 ≈ 22.47220505

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505 Logic

504 and Level 3

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Today’s Puzzle:

504 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-25

Something Cool about √504:

I’ve made a little cake to simplify √504:

504 cake

Just take the square root of everything on the outside of the cake and multiply them together: √504 = (√4)(√9)(√14) = (2 x 3)(√14) = 6√14 ≈ 22.449944, a pretty cool-looking approximation!

Factors of 504:

  • 504 is a composite number.
  • Prime factorization: 504 = 2 x 2 x 2 x 3 x 3 x 7, which can be written 504 = (2^3) x (3^2) x 7
  • The exponents in the prime factorization are 3, 2 and 1. Adding one to each and multiplying we get (3 + 1)(2 + 1)(1 + 1) = 4 x 3 x 2 = 24. Therefore 504 has exactly 24 factors.
  • Factors of 504: 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 18, 21, 24, 28, 36, 42, 56, 63, 72, 84, 126, 168, 252, 504
  • Factor pairs: 504 = 1 x 504, 2 x 252, 3 x 168, 4 x 126, 6 x 84, 7 x 72, 8 x 63, 9 x 56, 12 x 42, 14 x 36, 18 x 28 or 21 x 24
  • Taking the factor pair with the largest square number factor, we get √504 = (√36)(√14) = 6√14 ≈ 22.449944

Sum-Difference Puzzle:

504 has twelve factor pairs. One of the factor pairs adds up to 65, and a different one subtracts to 65. If you can identify those factor pairs, then you can solve this puzzle!

A Puzzle about the Number 504:

The numbers 360, 420, 480, and 504 have something in common? The first three numbers are all multiples of 60, but 504 isn’t. 504 is the smallest number that isn’t divisible by 60 that has the same special something that those other three numbers have. Can you figure out what it is? Hint: It has something to do with the number 24.

Factors for Today’s Puzzle:

504 Factors

503 and Level 2

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503 = (2^3) + (3^3) + (5^3) + (7^3) which is the sum of the cubes of the first four prime numbers. 503 is the smallest prime number that is the sum of consecutive cubes of prime numbers.

503 is also the sum of three consecutive prime numbers: 163, 167, and 173.

503 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-25

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  • 503 is a prime number.
  • Prime factorization: 503 is prime.
  • The exponent of prime number 503 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 503 has exactly 2 factors.
  • Factors of 503: 1, 503
  • Factor pairs: 503 = 1 x 503
  • 503 has no square factors that allow its square root to be simplified. √503 ≈ 22.42766

How do we know that 503 is a prime number? If 503 were not a prime number, then it would be divisible by at least one prime number less than or equal to √503 ≈ 22.4. Since 503 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 503 is a prime number.

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503 Factors