1128 is the 24th Hexagonal Number

1128 is the 24th hexagonal number because of the way that it can be factored:
2(24²) – 24 = 1128,
(2(24) – 1)24 = 1128
or simply 47(24) = 1128.

This is what the 24th hexagonal number looks like when it is made with 1128 tiny squares arranged into a hexagon:

All hexagonal numbers are also triangular numbers. 1128 is the 47th triangular number:

What else can the factors of 1128 tell us?

  • 1128 is a composite number.
  • Prime factorization: 1128 = 2 × 2 × 2 × 3 × 47, which can be written 1128 = 2³ × 3 × 47
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 1128 has exactly 16 factors.
  • Factors of 1128: 1, 2, 3, 4, 6, 8, 12, 24, 47, 94, 141, 188, 282, 376, 564, 1128
  • Factor pairs: 1128 = 1 × 1128, 2 × 564, 3 × 376, 4 × 282, 6 × 188, 8 × 141, 12 × 94, or 24 × 47
  • Taking the factor pair with the largest square number factor, we get √1128 = (√4)(√282) = 2√282 ≈ 33.58571

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What Kind of Shape Is 946 In?

First of all, 946 is the sum of the numbers from 1 to 43, so it is the 43rd triangular number.

Every other triangular number is also a hexagonal number. Since 946 is the 43rd triangular number, and 43 is an odd number, 946 is also the 22nd hexagonal number. 946 is the 22nd hexagonal number because 22(2(22) – 1) = 22(43) = 946.

But that’s not all. 946 is different than any previous hexagonal number. 946 is the smallest hexagonal number that is also a hexagonal pyramidal number. It is, in fact, the 11th hexagonal pyramidal number. That means if you stack the hexagons in the graphic below in order from largest to smallest, you would get a hexagonal pyramid made with 946 tiny squares. That’s pretty cool, I think.

 

467 + 479 = 946 so 946 is the sum of two consecutive prime numbers.
946 is also the sum of the twenty prime numbers from 11 to 89.

946 is palindrome 181 in BASE 27 because
1(27²) + 8(27¹) + 1(27⁰) = 729 + 216 + 1 = 946

  • 946 is a composite number.
  • Prime factorization: 946 = 2 × 11 × 43
  • 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 × 2 × 2 = 8. Therefore 946 has exactly 8 factors.
  • Factors of 946: 1, 2, 11, 22, 43, 86, 473, 946
  • Factor pairs: 946 = 1 × 946, 2 × 473, 11 × 86, or 22 × 43
  • 946 has no square factors that allow its square root to be simplified. √946 ≈ 30.75711