divisibility
Imagine handing out a bag of candies so that everyone gets the same whole number and nothing is left over. When that is possible, the number of candies is divisible by the number of people. Divisibility is simply the question of whether one whole number splits another into equal whole pieces with no scraps.
Formally, an integer a divides an integer b if there is an integer k with b = a times k. We write a | b, read “a divides b.” For example 3 | 12 because 12 = 3 times 4, but 3 does not divide 13 because no integer times 3 lands on 13. The divisor (here a) must be a whole number, and so must the quotient k.
Two honest cautions. First, every integer divides 0, since 0 = a times 0 for any a, but 0 divides only 0 itself — you cannot have b = 0 times k unless b is already 0. Second, divisibility is about exactness: 13 leaving remainder 1 when divided by 3 is not divisibility, it is just ordinary division with a remainder.
Does 7 divide 56? Yes: 56 = 7 times 8, so 7 | 56. Does 7 divide 50? No: 50 = 7 times 7 + 1, a remainder of 1 remains.
Divisibility means a remainder of exactly zero.
Handy divisibility rules: a number is divisible by 2 if its last digit is even, by 3 if its digit sum is divisible by 3, by 5 if it ends in 0 or 5, and by 9 if its digit sum is divisible by 9.