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12: It is divisible by 3 and by 4. [6] 324: it is divisible by 3 and by 4. Subtract the last digit from twice the rest. The result must be divisible by 12. 324: 32 × 2 − 4 = 60 = 5 × 12. 13: Form the alternating sum of blocks of three from right to left. The result must be divisible by 13. [7] 2,911,272: 272 − 911 + 2 = −637.
Every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100, but these centurial years are leap years if they are exactly divisible by 400. For example, the years 1700, 1800, and 1900 are not leap years, but the years 1600 and 2000 are.
A year divisible by 100 is not a leap year in the Gregorian calendar unless it is also divisible by 400. For example, 1600 was a leap year, but 1700, 1800 and 1900 were not. Some programs may have relied on the oversimplified rule that "a year divisible by four is a leap year".
In the past 500 years, there was no leap day in 1700, 1800 and 1900, but 2000 had one. In the next 500 years, if the practice is followed, there will be no leap day in 2100, 2200, 2300 and 2500 ...
In number theory, reversing the digits of a number n sometimes produces another number m that is divisible by n. This happens trivially when n is a palindromic number; the nontrivial reverse divisors are 1089, 2178, 10989, 21978, 109989, 219978, 1099989, 2199978, ... (sequence A008919 in the OEIS).
That is, although 360 and 2520 both have more divisors than any number twice themselves, 2520 is the lowest number divisible by both 1 to 9 and 1 to 10, whereas 360 is not the lowest number divisible by 1 to 6 (which 60 is) and is not divisible by 1 to 7 (which 420 is).
A century leap year is a leap year in the Gregorian calendar that is evenly divisible by 400. [1] Like all leap years, it has an extra day in February for a total of 366 days instead of 365. In the obsolete Julian calendar, all years that were divisible by 4, including end-of-century years, were considered leap years. The Julian rule, however ...
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