multiplication

637 and Level 2

/ ivasallay / Leave a comment

637 is the sum of two perfect squares, 441 and 196, so it is the hypotenuse of a Pythagorean triple, namely 245-588-637. The greatest common factor of those FIVE numbers is also a perfect square. What is it?

637 is the sum of the nineteen prime numbers from 3 to 71.

637 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-05

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  • 637 is a composite number.
  • Prime factorization: 637 = 7 x 7 x 13, which can be written 637 = (7^2) x 13
  • 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 637 has exactly 6 factors.
  • Factors of 637: 1, 7, 13, 49, 91, 637
  • Factor pairs: 637 = 1 x 637, 7 x 91, or 13 x 49
  • Taking the factor pair with the largest square number factor, we get √637 = (√49)(√13) = 7√13 ≈ 25.2388589.

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637 Factors

633 and Level 5

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  • 633 = 210 + 211 + 212 (three consecutive numbers)
  • 633 = 209 + 211 + 213 (three consecutive odd numbers)
  • 633 = 199 + 211 + 223 (three consecutive prime numbers)

Each of the sums above has 3 numbers, and 3 is a prime factor of 633. The middle number in each of the sums is 211 which is the other prime factor of 633.

Each of its two prime factors is 104 away from their average, 107.

Thus (107^2) – (104^2) = 633

The numbers in 633’s other factor pair are 1 and 633, and they are each 316 away from their average, 317.

Thus (317^2) – (316^2) = 633

633 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-28

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  • 633 is a composite number.
  • Prime factorization: 633 = 3 x 211
  • 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 633 has exactly 4 factors.
  • Factors of 633: 1, 3, 211, 633
  • Factor pairs: 633 = 1 x 633 or 3 x 211
  • 633 has no square factors that allow its square root to be simplified. √633 ≈ 25.15949.

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633 Logic

 

 

632 and Level 4

/ ivasallay / Leave a comment

Since 632 is greater than 120 and is divisible by 8 but not by 16, it can be written as the sum of 16 consecutive counting numbers:

32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 = 632

632 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-28

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  • 632 is a composite number.
  • Prime factorization: 632 = 2 x 2 x 2 x 79, which can be written 632 = (2^3) x 79
  • 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 632 has exactly 8 factors.
  • Factors of 632: 1, 2, 4, 8, 79, 158, 316, 632
  • Factor pairs: 632 = 1 x 632, 2 x 316, 4 x 158, or 8 x 79
  • Taking the factor pair with the largest square number factor, we get √632 = (√4)(√158) = 2√158 ≈ 25.13961.

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632 Logic

629 and Level 1

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629 is the sum of the 17 prime numbers from 7 to 71. Both of its prime factors, 17 and 37, are included in that list.

17 and 37 are both 10 numbers away from their average, 27. That means that 629 + 10² = 729 or 27².

25² + 2² = 629 and 23² + 10² = 629. Notice that 629 plus or minus 100 is a square number.

Both of 629’s prime factors have a remainder of one when divided by four so 629 is the hypotenuse of four Pythagorean triples, two of which are primitives.

  • 100-621-629, a primitive that reminds me of another primitive, 20-21-29
  • 204-595-629, three numbers whose greatest common factor is 17
  • 296-555-629, three numbers whose greatest common factor is 37
  • 429-460-629, a primitive whose shorter leg is exactly 200 less than its hypotenuse.

629 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-28

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  • 629 is a composite number.
  • Prime factorization: 629 = 17 x 37
  • 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 629 has exactly 4 factors.
  • Factors of 629: 1, 17, 37, 629
  • Factor pairs: 629 = 1 x 629 or 17 x 37
  • 629 has no square factors that allow its square root to be simplified. √629 ≈ 25.079872.

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629 Factors

616 and Level 2

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The Padovan sequence produces a lovely spiral of equilateral triangles similar to the spiral made from golden rectangles and the Fibonacci sequence.

The Padovan sequence begins with the following numbers: 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081 …

The first two numbers in the Fibonacci sequence are both 1’s. After that a number, n, in the Fibonacci sequence is found by adding together (n-2) and (n-1).

The first three numbers in the Padovan sequence are all 1’s. After that a number, n, in the Padovan sequence is found by adding together (n-3) and (n-2), and 616 is one of those numbers.

616 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-14

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  • 616 is a composite number.
  • Prime factorization: 616 = 2 x 2 x 2 x 7 x 11, which can be written 616 = (2^3) x 7 x 11
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 616 has exactly 16 factors.
  • Factors of 616: 1, 2, 4, 7, 8, 11, 14, 22, 28, 44, 56, 77, 88, 154, 308, 616
  • Factor pairs: 616 = 1 x 616, 2 x 308, 4 x 154, 7 x 88, 8 x 77, 11 x 56, 14 x 44, or 22 x 28
  • Taking the factor pair with the largest square number factor, we get √616 = (√4)(√154) = 2√154 ≈ 24.819347.

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616 Factors

 

612 and Level 5

/ ivasallay / 2 Comments

612 is the hypotenuse of the Pythagorean triple 288-540-612. Which factor of 612 is the greatest common factor of those three numbers?

612 = 17 x 36, which is 17 x 18 x 2, and that is exactly four times the formula of the 17th triangular number. Thus . . .

612 = 4(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17)

612 can be expressed as the sum of consecutive counting numbers in several ways:

  • 612 = 203 + 204 + 205 (3 consecutive numbers because it is divisible by 3)
  • 612 = 73 + 74 + 75 + 76 + 77 + 78 + 79 + 80 (eight consecutive numbers because it is divisible by 4, but not 8)
  • 612 = 64 + 65 + 66 + 67 + 68 + 69 + 70 + 71 + 72 (9 consecutive numbers because it is divisible by 9)
  • 612 = 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 (17 consecutive numbers because it is divisible by 17)
  • 612 = 14 + 15 + 16 + 17 + . . . . + 34 + 35 + 36 + 37 (24 consecutive numbers because 612 is divisible by 12, but not by 24)

What is a relationship between the numbers in bold print and the number 612?

612 is also the sum of the twelve prime numbers from 29 to 73.

612 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-07

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  • 612 is a composite number.
  • Prime factorization: 612 = 2 x 2 x 3 x 3 x 17, which can be written 612 = (2^2) x (3^2) x 17
  • The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 x 3 x 2 = 18. Therefore 612 has exactly 18 factors.
  • Factors of 612: 1, 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204, 306, 612
  • Factor pairs: 612 = 1 x 612, 2 x 306, 3 x 204, 4 x 153, 6 x 102, 9 x 68, 12 x 51, 17 x 36 or 18 x 34
  • Taking the factor pair with the largest square number factor, we get √612 = (√36)(√17) = 6√17 ≈ 24.73863

Although I prefer using a modified cake method to find square roots, most people prefer factor trees. If you use a factor tree, I suggest you still look for easy-to-detect perfect square factors (100, 4, 9, 25) so that the most common duplicate prime factors are together:

612 Factor tree

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612 Logic

610 and Level 3

/ ivasallay / Leave a comment

The first 17 Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597. Notice that 610 is the 15th Fibonacci number.

There is a fascinating relationship between some of the Fibonacci numbers and some of the Markov numbers. 610 is the 12th Markov number. Get out your calculator and satisfy yourself that the following two Diophantine equations involving Fibonacci/Markov numbers are true:

1² + 233² + 610² = 3(1)(233)(610)

1² + 610² + 1597² = 3(1)(610)(1597)

Here is a fascinating fact I learned from twitter:

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Starting with 5, every other Fibonacci number would be 5, 13, 34, 89, 233, 610, 1597, . . .

610 is on that list. What could be the integer sides of a right triangle with 610 as the hypotenuse?

There are actually FOUR such triangles, namely. . .

  • 110-600-610
  • 272-546-610
  • 414-448-610
  • 366-488-610

None of those are primitives, but it is a great list nonetheless!

610 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-07

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  • 610 is a composite number.
  • Prime factorization: 610 = 2 x 5 x 61
  • 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 610 has exactly 8 factors.
  • Factors of 610: 1, 2, 5, 10, 61, 122, 305, 610
  • Factor pairs: 610 = 1 x 610, 2 x 305, 5 x 122, or 10 x 61
  • 610 has no square factors that allow its square root to be simplified. √610 ≈ 24.698178.

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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.

610 Factors

609 and Level 2

/ ivasallay / Leave a comment

609 is the hypotenuse of the Pythagorean triple 420-441-609. What is the greatest common factor of those three numbers?

21 x 29 = 609, both odd numbers from a factor pair for 609, and (29 – 21)/2 = 4. So since 4^2 = 16, we are 16 counting numbers away from a perfect square, in fact the next perfect square. Written mathematically that is (25-4)(25 +4) = (25^2) – (4^2).

609 is the 11th strobogrammatic number which means it looks like the same number upside-down because it uses only the digits 0, 1, 6, 8, and 9.

There are other numbers that don’t look the same upside-down but look the same reflected in a mirror. Check out teachfuthermaths for a fun puzzle about some of those numbers.

 609 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-07

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  • 609 is a composite number.
  • Prime factorization: 609 = 3 x 7 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 609 has exactly 8 factors.
  • Factors of 609: 1, 3, 7, 21, 29, 87, 203, 609
  • Factor pairs: 609 = 1 x 609, 3 x 203, 7 x 87, or 21 x 29
  • 609 has no square factors that allow its square root to be simplified. √609 ≈ 24.677925.

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609 Factors

579 and Level 1

/ ivasallay / Leave a comment

579 is the hypotenuse of the Pythagorean triple 285-504-579. Which of 579’s factors is the greatest common factor of those three numbers?

Last week someone googled “find,the least 6 digit which has 173 factor” and arrived at Findthefactors.com. Here how to find the answer: Divide 100,000 by 173 and get 578.03 approximately. Round that answer up to 579. Multiply 579 by 173 and get 100167, the smallest 6-digit number that has 173 as a factor.

579 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-08-10

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  • 579 is a composite number.
  • Prime factorization: 579 = 3 x 193
  • 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 579 has exactly 4 factors.
  • Factors of 579: 1, 3, 193, 579
  • Factor pairs: 579 = 1 x 579 or 3 x 193
  • 579 has no square factors that allow its square root to be simplified. √579 ≈ 24.0624188

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579 Factors

576 and Level 5

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The number formed from the last 2 digits of 576 is divisible by 4. That means 576 is also divisible by 4.

576 is made from 3 consecutive numbers so it is divisible by 3. Since the middle number, 6, is also divisible by 3, we know that 576 is also divisible by 9.

Either of the previous statements is enough to indicate that √576 can be reduced. In fact, boy, can it ever be reduced!

576 is the smallest number to have exactly 21 factors. (Only perfect squares have an odd number of factors.)

576 is also the sum of consecutive primes in two different ways:

  • 283 + 293 = 576
  • 137 + 139 + 149 + 151 = 576

576 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-08-03

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  • 576 is a composite number.
  • Prime factorization: 576 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3, which can be written 576 = (2^6) x (3^2)
  • The exponents in the prime factorization are 6 and 2. Adding one to each and multiplying we get (6 + 1)(2 + 1) = 7 x 3 = 21. Therefore 576 has exactly 21 factors.
  • Factors of 576: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, 288, 576
  • Factor pairs: 576 = 1 x 576, 2 x 288, 3 x 192, 4 x 144, 6 x 96, 8 x 72, 9 x 64, 12 x 48, 16 x 36, 18 x 32 or 24 x 24
  • 576 is a perfect square. √576 = 24

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576 Logic