triangular number

526 is a Centered Pentagonal Number

/ ivasallay / 3 Comments

I am excited that Kerrydunton included the Find the Factors puzzles on a list of great maths starters.

1 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 + 55 + 60 + 65 + 70 = 526. Thus, 526 is the 15th centered pentagonal number.

The above equation is the same as saying that 526 is one more than five times the 14th triangular number.

526 Centered Pentagonal Number

The last digit of a centered pentagonal number is always 1 or 6. Triangular numbers greater than 3 are always composite numbers, but centered pentagonal numbers ending in 1 might be prime numbers. For example 31, 181, 331, 601, 1051, 1381, and 3331 are centered pentagonal numbers and prime numbers.

526 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-15

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  • 526 is a composite number.
  • Prime factorization: 526 = 2 x 263
  • 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 526 has exactly 4 factors.
  • Factors of 526: 1, 2, 263, 526
  • Factor pairs: 526 = 1 x 526 or 2 x 263
  • 526 has no square factors that allow its square root to be simplified. √526 ≈ 22.93468988

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526 Logic

351 is a Triangular Number

/ ivasallay / Leave a comment

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

210 and Level 4

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Today’s Puzzle:

 2014-33 Level 4

Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-08-18

Factors of 210:

  • 210 is a composite number and a triangular number.
  • Prime factorization: 210 = 2 x 3 x 5 x 7, so 210 is the product of the first four prime numbers!
  • 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 210 has 16 factors.
  • Factors of 210: 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210
  • Factor pairs: 210 = 1 x 210, 2 x 105, 3 x 70, 5 x 42, 6 x 35, 7 x 30, 10 x 21, or 14 x 15
  • Since 210 has no square factors, √210 cannot be simplified. √210 ≈ 14.491

Sum-Difference Puzzles:

210 has eight factor pairs. One of those factor pairs adds up to 29, and another one subtracts to 29. Another factor pair adds up to 37, and a different one subtracts to 37. Thus, 210 produces two different Sum-Difference Puzzles!

210 is the smallest number that makes sum-difference two different ways!

More about the Number 210:

210 is the hypotenuse of a Pythagorean triple:
126-168-210, which is (3-4-5) times 42.

Because 20 + 1 = 21 and (20/2) x 21 = 210, we know that 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 = 210, thus it is the 20th triangular number.

A Logical Way to Solve Today’s Puzzle:

2014-33 Level 4 Logic

171 and Level 4

/ ivasallay / Leave a comment
171 is a composite number.
Factor pairs: 171 = 1 x 171, 3 x 57, or 9 x 19
Factors of 171: 1, 3, 9, 19, 57, 171
Prime factorization: 170 = 3 x 3 x 19, which can be written 170 = (3^2) x 19

171 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 and is therefore a triangular number.

The fact that 171 is a triangular number is easy to recognize because 170 = 9 x 19 = (18/2) x 19. Any integer resulting from n(n + 1)/2 is a triangular number.

2014-27 Level 4

Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-07-07

2014-27 Level 4 Logic