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It is the third (or second) dodecahedral number, [4] and the sum of the first seven triangular numbers (1, 3, 6, 10, 15, 21, 28), which makes it the seventh tetrahedral number. [5] The number of divisors of 84 is 12. [6] As no smaller number has more than 12 divisors, 84 is a largely composite number. [7]
The table shown on the right can be used in a two-sample t-test to estimate the sample sizes of an experimental group and a control group that are of equal size, that is, the total number of individuals in the trial is twice that of the number given, and the desired significance level is 0.05. [4] The parameters used are:
231 is also a multiple of 3: one has 231 = 3 · 77, and thus n = 2 · 3 2 · 77. Continue with 77, and 3 as a first divisor candidate. 77 is not a multiple of 3, since the sum of its digits is 14, not a multiple of 3. It is also not a multiple of 5 because its last digit is 7. The next odd divisor to be tested is 7.
If a factor already has natural units, then those are used. For example, a shrimp aquaculture experiment [9] might have factors temperature at 25 °C and 35 °C, density at 80 or 160 shrimp/40 liters, and salinity at 10%, 25% and 40%. In many cases, though, the factor levels are simply categories, and the coding of levels is somewhat arbitrary.
The next number not yet crossed out in the list after 5 is 7; the next step would be to cross out every 7th number in the list after 7, but they are all already crossed out at this point, as these numbers (14, 21, 28) are also multiples of smaller primes because 7 × 7 is greater than 30.
In other words, for the quantities a and b, it can be said that b is a multiple of a if b = na for some integer n, which is called the multiplier. If a is not zero, this is equivalent to saying that / is an integer. When a and b are both integers, and b is a multiple of a, then a is called a divisor of b. One says also that a divides b.
Integer factorization is the process of determining which prime numbers divide a given positive integer.Doing this quickly has applications in cryptography.The difficulty depends on both the size and form of the number and its prime factors; it is currently very difficult to factorize large semiprimes (and, indeed, most numbers that have no small factors).
lcm(m, n) (least common multiple of m and n) is the product of all prime factors of m or n (with the largest multiplicity for m or n). gcd( m , n ) × lcm( m , n ) = m × n . Finding the prime factors is often harder than computing gcd and lcm using other algorithms which do not require known prime factorization.