PINKLETES
DIVISIBILITY
Key terms:
​
1. Divisor: A divisor is a number that can divide another number without leaving a remainder. In other words, if you have two numbers, A and B, and A can be divided by B without any remainder, then B is a divisor of A. For example, in the case of 15 and 3, 3 is a divisor of 15 because 15 ÷ 3 equals 4 with no remainder.
​
2. Multiple: A multiple is a number that can be obtained by multiplying a given number by an integer (a whole number). If you take a number, A, and multiply it by another number, B, and the result is C, then C is a multiple of A. For example, if you have 4 as the given number (A) and you multiply it by 3 (B), you get 12, so 12 is a multiple of 4.
​
3. Factor: A factor is a number that divides another number evenly, meaning it is a divisor of that number. Factors are essentially the divisors of a given number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because these are the numbers that can divide 12 without leaving a remainder.
​
Let a and b be integers where a DOES NOT equal 0. We can say that "a divides b" if there is such an integer n such that b = a x n.
Example: if a=3 and b=15, we can say that 15=3 x n, where n=5
if a|b we say that a is a factor/divisor of b and b is a multiple of a.
Example: if a=3 and b=15, we can say that 3 is a factor/divisor of 15 and 15 is a multiple of a
​
Rules: let a and b be integers.
​
1) a|0 for all integers Ex: 0 = 2 x 0
​
2) a|a (reflexivity) Ex: 2 = 1 x 2
​
3) if a|b and b|c, then a|c (transitivity) Ex: 2|4 and 4|8, 2|8
​
4) if a|b and a|c, then a |(bx±cy) for any integers x and y. Ex: 2|4 and 2|10, 2|(2x+10)
​
5) if a|b and a|(b±c), then a|c Ex: 2|10 and 2|(10-6), then 2|6
​
6) if a|b and b≠0, then |a| ≤ |b| Ex: -2|6, then |-2| ≤ |6|
​
7) if a|b and b|a, then a = ±b Ex: 4|-4 and -4|4, then 4 = -(-4)
DIVISIBILITY RULES 2-12
2: ​​divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
Ex: 24, 36, 70, 3248, 2024
​
3: divisible by 3 if the sum of its digits is divisible by 3.
Ex: 303, 21, 27,
​
4: A number is divisible by 4 if the last two digits form a number divisible by 4.
Ex: 116, 88, 128
​
5: A number is divisible by 5 if its last digit is 0 or 5.​
Ex: 230, 55, 115
​
6: A number is divisible by 6 if it is divisible by both 2 and 3.
Ex: 108 -> 1+8 = 9, which is divisible by 3, and 108 ends in 8
252 -> 2+5+2=9, which is divisible by 3, and 252 ends in 2
96 -> 9+6=15, which is divisible by 3, and 96 ends in 6
​
7: To check divisibility by 7, take the last digit, double it, and subtract the result from the remaining number. If the result is divisible by 7 (including 0), then the original number is divisible by 7. Repeat the process if necessary.
Ex: 441 -> 1(2) -> 44-2=42
42 is divisible by 7 so 441 is divisible by 7
​
8: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
Ex: 325112 -> 112 is divisible by 8
1104 -> 104 is divisible by 8
​
9: A number is divisible by 9 if the sum of its digits is divisible by 9. (similar rule for 3)
Ex: 918, 108, 999
​
10: A number is divisible by 10 if it ends in 0.
Ex, 50, 12300, 2020
​
11: To check divisibility by 11, alternate the difference and sum of the digits. If the result equals 11 or 1, then the original number is divisible by 11.
Ex: 1353 -> 1-3+5-3 = 0
99 -> 9-9=0
506 -> 5-0+6=11
​
12: A number is divisible by 12 if it is divisible by both 3 and 4.
Ex: 144 -> 144/3=48 and 144/4=36
60 -> 60/3=20 and 60/4 = 15
408 -> 408/3=136 and 408/4=102
​
​