1325 Hockey Stick

 

If someone you know loves hockey and wants a fun way to practice multiplication facts, this hockey stick could be the perfect gift.

Print the puzzles or type the solution in this excel file:10-factors-1321-1332

  • 1325 is a composite number.
  • Prime factorization: 1325 = 5 × 5 × 53, which can be written 1325 = 5² × 53
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 1325 has exactly 6 factors.
  • Factors of 1325: 1, 5, 25, 53, 265, 1325
  • Factor pairs: 1325 = 1 × 1325, 5 × 265, or 25 × 53
  • Taking the factor pair with the largest square number factor, we get √1325 = (√25)(√53) = 5√53 ≈ 36.40055

1325 is the sum of two squares in three different ways:
29² + 22² = 1325 
34² + 13² = 1325
35² + 10² = 1325

1325 is the hypotenuse of SEVEN Pythagorean triples:
115-1320-1325 which is 5 times (23-264-265)
357-1276-1325 calculated from 29² – 22², 2(29)(22), 29² + 22²
371-1272-1325 which is (7-24-25) times 53
480-1235-1325 which is 5 times (96-247-265)
700-1125-1325 calculated from 2(35)(10), 35² – 10², 35² + 10²
795-1060-1325 which is (3-4-5) times 265
884-987-1325 calculated from 2(34)(13), 34² – 13², 34² + 13²

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331 and Hockey Sticks

  • 331 is a prime number.
  • Prime factorization: 331 is prime.
  • The exponent of prime number 331 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 331 has exactly 2 factors.
  • Factors of 331: 1, 331
  • Factor pairs: 331 = 1 x 331
  • 331 has no square factors that allow its square root to be simplified. √331 ≈ 18.193

How do we know that 331 is a prime number? If 331 were not a prime number, then it would be divisible by at least one prime number less than or equal to √331 ≈ 18.193. Since 331 cannot be divided evenly by 2, 3, 5, 7, 11, 13, or 17, we know that 331 is a prime number.

2014-51 Level 3

 Print the puzzles or type the factors on this excel file:  10 Factors 2014-12-22

Today’s puzzle looks like a hockey stick. Hockey sticks remind me not only of the obvious winter sport but also of the Twelve Days of Christmas and Pascal’s triangle.

Dimacs.rutgers.edu explains quite nicely how a hockey stick in Pascal’s triangle can give you the total number of gifts received after one day, two days, three days, and so on. Look at the green and red hockey stick with bold black numbers in this illustration of Pascal’s triangle:

If someone gave you one partridge every day for 12 days, two turtle doves every day for 11 days, three French hens every day for 10 days, etc, etc, and etc, then you would receive 364 gifts. (364 is so easy to remember because it is one less than the number of days in a year.)

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.
2014-51 Level 3 Factors