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The basic rule for divisibility by 4 is that if the number formed by the last two digits in a number is divisible by 4, the original number is divisible by 4; [2] [3] this is because 100 is divisible by 4 and so adding hundreds, thousands, etc. is simply adding another number that is divisible by 4. If any number ends in a two digit number that ...
Plot of the number of divisors of integers from 1 to 1000. Highly composite numbers are in bold and superior highly composite numbers are starred. ... 3, 5, 7, 9, 15 ...
2005 – A vertically symmetric number; 2006 – number of subsets of {1,2,3,4,5,6,7,8,9,10,11} with relatively prime elements [6] 2007 – 2 2007 + 2007 2 is prime [7] 2008 – number of 4 × 4 matrices with nonnegative integer entries and row and column sums equal to 3 [8] 2009 = 7 4 − 7 3 − 7 2; 2010 – number of compositions of 12 into ...
Later, on a calendar yet to come (we'll get to it), it was decreed that years divisible by 100 not follow the four-year leap day rule un ... 1800 and 1900, but 2000 had one. In the next 500 years ...
the k given prime numbers p i must be precisely the first k prime numbers (2, 3, 5, ...); if not, we could replace one of the given primes by a smaller prime, and thus obtain a smaller number than n with the same number of divisors (for instance 10 = 2 × 5 may be replaced with 6 = 2 × 3; both have four divisors);
This eliminates 3 of the 4 end-of-century years in a 400-year period. For example, the years 1600, 2000, 2400, and 2800 are century leap years since those numbers are evenly divisible by 400, while 1700, 1800, 1900, 2100, 2200, 2300, 2500, 2600, 2700, 2900, and 3000 are common years despite being evenly divisible by 4. This scheme brings the ...
A natural number is divisible by three if the sum of its digits in base 10 is divisible by 3. For example, the number 21 is divisible by three (3 times 7) and the sum of its digits is 2 + 1 = 3. Because of this, the reverse of any number that is divisible by three (or indeed, any permutation of its digits) is also divisible by three. For ...
It is not known whether an even-odd pair of amicable numbers exists, but if it does, the even number must either be a square number or twice one, and the odd number must be a square number. However, amicable numbers where the two members have different smallest prime factors do exist: there are seven such pairs known. [8]