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The first: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600 (sequence A000142 in the OEIS). 0! = 1 is sometimes included. A k-smooth number (for a natural number k) has its prime factors ≤ k (so it is also j-smooth for any j > k). m is smoother than n if the largest prime factor of m is below the largest of n.
The value of π as used by the Greek mathematician and scientist Ptolemy was 3;8,30 = 3 + 8 / 60 + 30 / 60 2 = 377 / 120 ≈ 3.141 666.... [28] Jamshīd al-Kāshī , a 15th-century Persian mathematician, calculated 2 π as a sexagesimal expression to its correct value when rounded to nine subdigits (thus to 1 / 60 9 ...
The sum of its factors (including one and itself) sum to 360, exactly three times 120. Perfect numbers are order two ( 2-perfect ) by the same definition. 120 is the sum of a twin prime pair (59 + 61) and the sum of four consecutive prime numbers (23 + 29 + 31 + 37), four consecutive powers of two (8 + 16 + 32 + 64), and four consecutive powers ...
In the second step, they were divided by 3. The final result, 4 / 3 , is an irreducible fraction because 4 and 3 have no common factors other than 1. The original fraction could have also been reduced in a single step by using the greatest common divisor of 90 and 120, which is 30. As 120 ÷ 30 = 4, and 90 ÷ 30 = 3, one gets
If none of its prime factors are repeated, it is called squarefree. (All prime numbers and 1 are squarefree.) For example, 72 = 2 3 × 3 2, all the prime factors are repeated, so 72 is a powerful number. 42 = 2 × 3 × 7, none of the prime factors are repeated, so 42 is squarefree. Euler diagram of numbers under 100:
≡ 60 light-minutes ≡ 1.079 252 8488 × 10 12 m: light-minute: ≡ 60 light-seconds ≡ 1.798 754 748 × 10 10 m: light-second: ≡ Distance light travels in one second in vacuum ≡ 299 792 458 m: light-year: ly ≡ Distance light travels in vacuum in 365.25 days [6] ≡ 9.460 730 472 5808 × 10 15 m: line: ln ≡ 1 ⁄ 12 in [7] = 0.002 11 ...
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The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b).. In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
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