What 1331 in Pascal’s Triangle Means

1331 is the third row of Pascal’s triangle. What does that mean?
It means that 11³ = 1331,
and it means that 1 + 3 + 3 + 1 = 8 = 2³
Those third powers are not a coincidence.

It means that (x + y)³ = 1x³yº + 3x²y¹ + 3x¹y² + 1xºy³ or written more simply,
(x + y)³ = x³ + 3x²y¹ + 3x¹y² + y³

Likewise, (x – y)³ = 1x³yº – 3x²y¹ + 3x¹y² – 1xºy³ or simply
(x – y)³ = x³ – 3x²y¹ + 3x¹y² – y³

It also means that if you flip a coin three times, you’ll get
three heads and no tails 1 way: (HHH),
two head and one tail 3 ways: (HHT); (HTH); (THH),
one head and two tails 3 ways: (HTT); (THT); (TTH), and
no heads and three tails 1 way: (TTT).

That’s just some of what that third row being 1331 means but here are a few more facts about the number 1331.

  • 1331 is a composite number.
  • Prime factorization: 1331 = 11 × 11 × 11, which can be written 1331 = 11³
  • The exponent in the prime factorization is 3. Adding one, we get (3 + 1) = 4. Therefore 1331 has exactly 4 factors.
  • Factors of 1331: 1, 11, 121, 1331
  • Factor pairs: 1331 = 1 × 1331 or 11 × 121
  • Taking the factor pair with the largest square number factor, we get √1331 = (√121)(√11) = 11√11 ≈ 36.48287
  • 1331 is a perfect cube.

1331 is cool in some other bases, too:
It is 1000 in BASE 11, and
it’s 131 in BASE 35

Advertisements

Applying Divisibility Tricks to 924

Let’s apply some basic divisibility tricks to find some of the factors of 924:

  1. Like every other counting number, 924 is divisible by 1.
  2. Since 924 is even, it is divisible by 2.
  3. 9 + 2 + 4 = 15, a number divisible by 3, so 924 is divisible by 3.
  4. Its last two digits, 24, is divisible by 4, so 924 is divisible by 4.
  5. Its last digit isn’t 0 or 5, so 924 is NOT divisible by 5.
  6. 924 is even and divisible by 3, so it is also divisible by 6.
  7. Since 92-2(4) = 84, a number divisible by 7, we know that 924 is also divisible by 7.
  8. Since its last two digits are divisible by 8, and the third to the last digit, 9, is odd, 924 is NOT divisible 8.
  9. 9 + 2 + 4 = 15, a number not divisible by 9, so 924 is NOT divisible by 9.
  10. The last digit is not 0, so 924 is NOT divisible by 10.
  11. 9 – 2 + 4 = 11, so 924 is divisible by 11.

Thus, 1, 2, 3, 4, 6, 7, and 11 are all factors of 924. You can see its prime factors easily on the outside of its prime factor cake:

You may have seen one of its many possible factor trees contained in the first frame of this factor tree for 852,852 from my previous post:

924 is in the very center of the 12th row of Pascal’s triangle because 12!/(6!6!) = 924.

924 is the sum of consecutive prime numbers: 461 + 463 = 924

I like 924 written in some other bases:

770 BASE 11

336 BASE 17

220 BASE 21

SS BASE 32, S is 28

S0 BASE 33

  • 924 is a composite number.
  • Prime factorization: 924 = 2 x 2 x 3 x 7 x 11, which can be written 924 = 2² x 3 x 7 x 11
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 x 2 = 24. Therefore 924 has exactly 24 factors.
  • Factors of 924: 1, 2, 3, 4, 6, 7, 11, 12, 14, 21, 22, 28, 33, 42, 44, 66, 77, 84, 132, 154, 231, 308, 462, 924
  • Factor pairs: 924 = 1 x 924, 2 x 462, 3 x 308, 4 x 231, 6 x 154, 7 x 132, 11 x 84, 12 x 77, 14 x 66, 21 x 44, 22 x 42, or 28 x 33,
  • Taking the factor pair with the largest square number factor, we get √924 = (√4)(√231) = 2√231 ≈ 30.3973683.

924 has several sets of consecutive factors. Besides being divisible by the 1st, 2nd, and 3rd triangular numbers (1, 3, and 6), those consecutive factors mean the following:

  • 924 is divisible by the 6th triangular number, 21, which is 6(7)/2.
  • 924 is divisible by the 11th triangular number, 66, which is 11(12)/2.
  • 924 is divisible by the 21st triangular number, 231, which is 21(22)/2.