Wikipedia:Reference desk/Archives/Mathematics/2018 January 18

= January 18 =

Relation between multiples of different numbers
How can the divisibility to number, say 7, be expressed based on divisibility to a lesser number like 3 or 5? Is there a general solution for numbers a greater than b? (Thanks!)--5.2.200.163 (talk) 15:26, 18 January 2018 (UTC)
 * See Least common multiple.  As 3, 5 and 7 are prime numbers the only numbers which are divisible by two of them are multiples of their product. 195.147.104.148 (talk)  —Preceding undated comment added 15:59, 18 January 2018 (UTC)
 * In the general case, where the numbers aren't all prime, there are some rules you can apply, which are nicely summarized here: http://www.mathwarehouse.com/arithmetic/numbers/divisibility-rules-and-tests.php#divisibilityBy8. Some of these rules are of the form you are looking for: for example, if a number is divisible by 6, it must be divisible by 2 and 3. You can extend this so that any composite number (i.e. a number that isn't prime) is divisible by any of its prime factors, and by any other composite number derived from those factors: for example, 60 has the factors 2,2,3, and 5, so it is divisible by 3 and 5, but also by 15, which is 3 * 5, by 6 (2*3), by 4 (2*2), and by 12 (2*2*3, or if you prefer, 4*3).OldTimeNESter (talk) 04:44, 19 January 2018 (UTC)
 * No need to send people elsewhere, we've got our own divisibility rule article. Rojomoke (talk) 14:57, 19 January 2018 (UTC)

I have posted this topic because of encountering somewhere (perhaps here) a statement involving the relation between divisibility to 9 and the status of being a perfect cube. The statement says that all perfect cubes are either multiples of 9 or 1 more or less than a multiple of 9. Therefore I wonder what methods are to sieve the multiples of a number a>b in regards to the multiples of a number b smaller than it, in this sense I intended the initial post.--5.2.200.163 (talk) 16:49, 23 January 2018 (UTC)


 * Every third integer is a multiple of 3, so any given integer is either a multiple of 3 or one more or one less than a multiple of 3. If x is a multiple of 3, then x = 3a for some integer a, and x3 = (3a)3 = 27a3 = 9⋅3a3 and is thus a multiple of 9.  If x is one more or one less than a multiple of 3, then x = 3a±1 for some integer a, and x3 = (3a±1)3 = 27a3±3⋅9a2+3⋅3a±1 = 9(3a3±3a2+a)±1 and is thus one more or one less than a multiple of 9.
 * Are you asking if a more general rule exists, relating perfect nth powers to a certain proximity of multiples of n2? Well, your perfect cube relation took advantage of the binomial coefficient of the penultimate term being C(3,1) = 3.  You still have that to work with, as the binomial coefficient of the penultimate term of (na+k)n is C(n,1) = n, so all terms but the very last will be multiples of n2.  But the problem comes from k no longer being restricted to 0 or ±1.  For a given natural number n, every integer x can be written as x = na + k for some integers a and k with -(n-1)/2 ≤ k ≤ (n-1)/2 for n odd and -n/2 < k ≤ n/2 for n even (as we did when n = 3 and we chose -1 ≤ k ≤ 1).  Then the final term of (na+k)n (the only one not necessarily a multiple of n2) is kn, but since k is no longer restricted to 0 or ±1, neither is kn.  Still, there are n-specific rules.
 * For n = 2, k is 0 or 1, so all perfect squares are either a multiple of 4 or are one more than a multiple of 4.
 * For n = 4, -1 ≤ k< 2, (±1)4 = 1, and 24 = 16, so all perfect fourth powers a multiple of 16 or one more than a multiple of 16.
 * For n = 5, -2 ≤ k< 2, (±1)5 = ±1, and (±2)5 = ±32 ≡ ±7 (mod 25), so all perfect fifth powers are a multiple of 25, one more or one less than a multiple 25, or 7 more or 7 less than a multiple of 25.
 * And so on. Not much of a sieve, I don't think. -- ToE 00:26, 24 January 2018 (UTC)


 * 1: 0
 * 2: 0,  1
 * 3: 0, ±1
 * 4: 0,  1,    0
 * 5: 0, ±1,   ±7
 * 6: 0,  1,   -8, 9
 * 7: 0, ±1,  ∓19, ∓18
 * 8: 0,  1,    0, -31,   0
 * 9: 0, ±1,  ±26,   0, ±28
 * 10: 0, 1,   24,  49, -24,   25
 * 11: 0, ±1,  ∓9,  ±3, ∓40,  ±27
 * 12: 0, 1,   64, -63,  64,    1,   0
 * 13: 0, ±1, ±80, ∓23, ∓22,  ±70, ±19
 * 14: 0, 1,  -80, -19, -68,  -31, -48,   49
 * 15: 0, ±1, ∓82, ∓18, ∓26, ∓100, ∓99, ∓107
 * 16: 0, 1,    0,  65,   0,  -63,    0, -127,    0
 * 17: 0, ±1, ∓134, ∓65, ±38, ∓131, ±40,  ±75, ±110
 * 18: 0, 1,   28,  81, 136, ∓107,    0,  109,  -80, 81
 * 19: 0, ±1, ±116, ∓54, ±99, ±62, ∓127,  ∓69,  ∓68, 28
 * 20: 0, 1,  176,   1, 176, -175,  176,    1,  176,  1, 0