547 and Level 4

2015-07-092017-07-29 / ivasallay / Leave a comment

547 is the difference of two consecutive numbers that have been cubed.

Thus 547 = 14³ – 13³.

We can use divisibility tricks that easily show that 547 is not divisible by 2, 3, 5, or 11.

547 is odd so it’s not divisible by 2
5 + 4 + 7 = 16 which is not divisible by 3, so 547 is not divisible by 3
The last digit of 547 is not 5 or 0 so it is not divisible by 5
5 – 4 + 7 = 8 which is not divisible by 11, so 547 is not divisible by 11

Also because it is the difference of 14^3 and 13^3, and 13 and 14 have no common prime factors, 547 also cannot be evenly divided by 13, 14 or any of their prime factors (2 and 7). Thus 547 is not divisible by 2, 3, 5, 7, 11, or 13. Could 547 possibly be a prime number? We only have to try to divide it by 17, 19, and 23 to know for sure! Scroll down past the puzzle to find out.

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-06

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547 is a prime number.
Prime factorization: 547 is prime.
The exponent of prime number 547 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 547 has exactly 2 factors.
Factors of 547: 1, 547
Factor pairs: 547 = 1 x 547
547 has no square factors that allow its square root to be simplified. √547 ≈ 23.38803

How do we know that 547 is a prime number? If 547 were not a prime number, then it would be divisible by at least one prime number less than or equal to √547 ≈ 23.4. Since 547 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 547 is a prime number.

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546 and Level 3

2015-07-082021-02-03 / ivasallay / Leave a comment

A Math Vocabulary Puzzle:

Puzzles can be a great resource in teaching mathematics. I’ve made crossword puzzles to review vocabulary words before, but none of them looked as inviting to complete as the one Resourceaholic recommended in her 7 July 2015 post on end of term resources. What a difference a little clip art makes to a well-constructed crossword puzzle! The post also includes links to several other amazing mathematics-related puzzles including a polygon word search and a number sequence number search puzzle. Check it out!

Today’s Puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-06

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.

Factors of 546:

546 is a composite number.
Prime factorization: 546 = 2 x 3 x 7 x 13
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 546 has exactly 16 factors.
Factors of 546: 1, 2, 3, 6, 7, 13, 14, 21, 26, 39, 42, 78, 91, 182, 273, 546
Factor pairs: 546 = 1 x 546, 2 x 273, 3 x 182, 6 x 91, 7 x 78, 13 x 42, 14 x 39, or 21 x 26
546 has no square factors that allow its square root to be simplified. √546 ≈ 23.36664

Sum-Difference Puzzle:

546 has eight factor pairs. One of the factor pairs adds up to 85, and a different one subtracts to 85. If you can identify those factor pairs, then you can solve this puzzle!

More about the Number 546:

546 is made from 3 consecutive numbers so it is divisible by 3.

The sum of all the prime numbers from 53 to 83 is 546. Can you list all those prime numbers?

546 is the hypotenuse of the Pythagorean triple 210-504-546. Those three numbers have eight common factors, but what is the greatest common factor?

545 and Level 2

2015-07-072017-03-11 / ivasallay / 3 Comments

The first few centered square numbers are 1, 5, 13, 25, 41, and 61. Starting in the center of this multi-colored square, can you locate each of those centered square numbers? 545 is the 17th centered square number. This wikipedia link explains the relationship between centered square numbers and the more familiar square numbers like 1, 4, 9, 16, 25 and 36. It also explains that every centered square number except 1 is the hypotenuse of a Pythagorean triple.

If you have difficulty seeing those first few centered square numbers, perhaps this will help:

545 is a centered square number because 16 and 17 are consecutive numbers and (16^2) + (17^2) = 545.

It is probably less exciting that (23^2) + (4^2) = 545.

545 is the hypotenuse of four Pythagorean triples. Which of these triples are primitives and which of them aren’t? The ones with greatest common factors greater than one are not primitive:

33-544-545
184-513-545
300-455-545
327-436-545

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-06

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545 is a composite number.
Prime factorization: 545 = 5 x 109
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 545 has exactly 4 factors.
Factors of 545: 1, 5, 109, 545
Factor pairs: 545 = 1 x 545 or 5 x 109
545 has no square factors that allow its square root to be simplified. √545 ≈ 23.345235

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544 and Level 1

2015-07-062017-03-11 / ivasallay / Leave a comment

544 is the hypotenuse of Pythagorean triple 256-480-544. Can you find the greatest common factor of those three numbers?

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-06

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544 is a composite number.
Prime factorization: 544 = 2 x 2 x 2 x 2 x 2 x 17, which can be written 544 = (2^5) x 17
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 544 has exactly 12 factors.
Factors of 544: 1, 2, 4, 8, 16, 17, 32, 34, 68, 136, 272, 544
Factor pairs: 544 = 1 x 544, 2 x 272, 4 x 136, 8 x 68, 16 x 34, or 17 x 32
Taking the factor pair with the largest square number factor, we get √544 = (√16)(√34) = 4√34 ≈ 23.3238

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542 and Level 6

2015-07-042017-03-11 / ivasallay / Leave a comment

Here are some ways to make 542 by adding together exactly three square numbers.

542 = (21^2) + (10^2) + (1^2)
542 = (19^2) + (10^2) + (9^2)
542 = (18^2) + (13^2) + (7^2)
542 = (15^2) + (14^2) + (11^2)

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-29

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542 is a composite number.
Prime factorization: 542 = 2 x 271
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 542 has exactly 4 factors.
Factors of 542: 1, 2, 271, 542
Factor pairs: 542 = 1 x 542 or 2 x 271
542 has no square factors that allow its square root to be simplified. √542 ≈ 23.280893

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541 and Level 5

2015-07-032017-07-29 / ivasallay / Leave a comment

541 = 21² + 10² = 441 + 100

541 is the hypotenuse of the primitive Pythagorean triple 341-420-541

And finally, after the longest string of composite numbers so far, 541 is quite notably a prime number. In fact, it is the 100th prime number.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-29

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541 is a prime number.
Prime factorization: 541 is prime.
The exponent of prime number 541 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 541 has exactly 2 factors.
Factors of 541: 1, 541
Factor pairs: 541 = 1 x 541
541 has no square factors that allow its square root to be simplified. √541 ≈ 23.25940 66 99 22 601 44

How do we know that 541 is a prime number? If 541 were not a prime number, then it would be divisible by at least one prime number less than or equal to √541 ≈ 23.3. Since 541 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 541 is a prime number.

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540 and Level 4

2015-07-022021-02-02 / ivasallay / 2 Comments

Today’s Puzzle:

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-29

Here is a logical order to use the clues to solve the puzzle:

Factors of 540:

540 is a composite number.
Prime factorization: 540 = 2 x 2 x 3 x 3 x 3 x 5, which can be written 540 = (2^2) x (3^3) x 5
The exponents in the prime factorization are 2, 3 and 1. Adding one to each and multiplying we get (2 + 1)(3 + 1)(1 + 1) = 3 x 4 x 2 = 24. Therefore 540 has exactly 24 factors.
Factors of 540: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, 270, 540
Factor pairs: 540 = 1 x 540, 2 x 270, 3 x 180, 4 x 135, 5 x 108, 6 x 90, 9 x 60, 10 x 54, 12 x 45, 15 x 36, 18 x 30 or 20 x 27
Taking the factor pair with the largest square number factor, we get √540 = (√36)(√15) = 6√15 ≈ 23.237900077

Sum-Difference Puzzles:

60 has six factor pairs. One of those pairs adds up to 17, and another one subtracts to 17. Put the factors in the appropriate boxes in the first puzzle.

540 has twelve factor pairs. One of the factor pairs adds up to 51, and a different one subtracts to 51. If you can identify those factor pairs, then you can solve the second puzzle!

The second puzzle is really just the first puzzle in disguise. Why would I say that?

More about the Number 540:

540 is the sum of the fourteen consecutive prime numbers from 13 to 67. Can you list all those prime numbers? It is also the sum of consecutive prime numbers 269 and 271.

540 has the same number of factors as 504. Both of those numbers tie with 360, 420, and 480 for the most factors so far.

540 is the hypotenuse of the Pythagorean triple 324-432-540. What is the greatest common factor of those three numbers?

540 is also an untouchable number.

The sum of the interior angles of every convex pentagon total 540 degrees.

Simplifying √539 and Level 3

2015-07-012019-08-14 / ivasallay / Leave a comment

If you add up all the prime numbers from 29 to 71, the sum will be 539.

Specifically, 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 = 539.

How to reduce √539: Since 539 cannot be evenly divided by 100, 4, 9, or 25, look for the smallest prime number that will divide into 539. You will soon note that 539 ÷ 7 = 77. Divide 77 by 7 again to get 11. I like to make a little cake that looks like this:

Then I take the square root of everything on the outside of the cake: √539 = (√7)(√7)(√11) = 7√11

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-29

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539 is a composite number.
Prime factorization: 539 = 7 x 7 x 11, which can be written 539 = (7^2) x 11
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 539 has exactly 6 factors.
Factors of 539: 1, 7, 11, 49, 77, 539
Factor pairs: 539 = 1 x 539, 7 x 77, or 11 x 49
Taking the factor pair with the largest square number factor, we get √539 = (√49)(√11) = 7√11 ≈ 23.21637353

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538 and Level 2

2015-06-302017-03-11 / ivasallay / Leave a comment

538 is the hypotenuse of the Pythagorean triple 138-520-538. Can you find the greatest common factor of those three numbers?

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-29

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538 is a composite number.
Prime factorization: 538 = 2 x 269
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 538 has exactly 4 factors.
Factors of 538: 1, 2, 269, 538
Factor pairs: 538 = 1 x 538 or 2 x 269
538 has no square factors that allow its square root to be simplified. √538 ≈ 23.194827

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537 and Level 1

2015-06-292017-03-11 / ivasallay / Leave a comment

537 is made from 3 consecutive odd numbers so it is divisible by 3.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-29

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537 is a composite number.
Prime factorization: 537 = 3 x 179
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 537 has exactly 4 factors.
Factors of 537: 1, 3, 179, 537
Factor pairs: 537 = 1 x 537 or 3 x 179
537 has no square factors that allow its square root to be simplified. √537 ≈ 23.17326

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Find the Factors

A Multiplication Based Logic Puzzle

number puzzle

547 and Level 4