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Similarly a number of the form 10x + y is divisible by 7 if and only if x + 5y is divisible by 7. [8] So add five times the last digit to the number formed by the remaining digits, and continue to do this until a number is obtained for which it is known whether it is divisible by 7. [9] Another method is multiplication by 3.
The numbers 1 and −1 are the only integers coprime with every integer, and they are the only integers that are coprime with 0. A number of conditions are equivalent to a and b being coprime: No prime number divides both a and b. There exist integers x, y such that ax + by = 1 (see Bézout's identity).
d() is the number of positive divisors of n, including 1 and n itself; σ() is the sum of the positive divisors of n, including 1 and n itselfs() is the sum of the proper divisors of n, including 1 but not n itself; that is, s(n) = σ(n) − n
If a is not divisible by p, that is, if a is coprime to p, then Fermat's little theorem is equivalent to the statement that a p − 1 − 1 is an integer multiple of p, or in symbols: [1] [2] (). For example, if a = 2 and p = 7 , then 2 6 = 64 , and 64 − 1 = 63 = 7 × 9 is a multiple of 7 .
A number x 0 is said to be a fixed point of a function f(x) if f(x 0) = x 0; in other words, if f leaves x 0 fixed. The fixed points of a function can be easily found graphically: they are simply the x coordinates of the points where the graph of f ( x ) intersects the graph of the line y = x .
Zelinsky proved that no three consecutive integers can all be refactorable. [1] Colton proved that no refactorable number is perfect . The equation gcd ( n , x ) = τ ( n ) {\displaystyle \gcd(n,x)=\tau (n)} has solutions only if n {\displaystyle n} is a refactorable number, where gcd {\displaystyle \gcd } is the greatest common divisor function.
85 is: the product of two prime numbers (5 and 17), and is therefore a semiprime of the form (5.q) where q is prime. specifically, the 24th Semiprime, it being the fourth of the form (5.q). together with 86 and 87, forms the second cluster of three consecutive semiprimes; the first comprising 33, 34, 35. [1]
They are successive lucky primes and sexy primes, both twice over, [4] [5] [6] and successive Pierpont primes, respectively the 9th and 8th. [ 7 ] 73 and 37 are consecutive values of g ( k ) {\displaystyle g(k)} such that every positive integer can be written as the sum of 73 or fewer sixth powers, or 37 or fewer fifth powers (and 19 or fewer ...