# Divisibility Rules

In general it is hard to determine from first glance whether one number is divisible by another. We need to perform division.

However, in some cases, we can determine whether one number divides another without dividing them by applying divisibility rules.

Divisibility of Sum. If some numbers are divisible by ${a}$ then their sum is also divisible by ${a}$.

If all addends except one are divisible by ${a}$ then sum is not divisible by ${a}$.

Example 1. Determine whether 55 is divisible by 11.

22 is divisible by 11 and 33 divisible by 11, therefore, 22+33=55 is divisible by 11.

Next example.

Example 2. Determine whether 126 is divisible by 5.

100 is divisible by 5, 20 is divisible by 5, 6 is not divisible by 5. Since exactly one addend is not divisible by 5, then their sum 100+20+6=126 is not divisible by 5.

Caution. If more than one addend is not divisible by ${a}$, we can't say that all sum is not divisible by ${a}$.

For example, 37 is not divisible by 4 and 19 is not divisible by 4, however 39+17=56 is divisible by 4.

Divisibility of Product. If at least one of numbers is divisible by ${a}$ then their product is also divisible by ${a}$.

Example 3. Determine whether 90 is divisible by 15.

Since 30 is divisible by 15 and ${90}={30}\cdot{3}$ then 90 is divisible by 15.

Next example.

Example 4. Determine whether 21 is divisible by 5.

Since neither 7 nor 3 are divisible by 5 then ${21}={7}\cdot{3}$ is not divisible by 5.

Divisibility of Product (2nd version). If ${a}$ is divisible by ${b}$ and ${c}$ is divisible by ${d}$ then ${a}\cdot{b}$ is divisible by ${c}\cdot{d}$.

For example, since 27 is divisible by 9 and 20 is divisible by 2 then ${27}\cdot{20}={540}$ is divisible by ${9}\cdot{2}={18}$.

Divisibility by 2. Number is even (divisible by 2) if and only if its last digit is even.

For example, 26 is divisible by 2 (6 is divisible by 2), while 43 is not divisible by 2 (3 is not divisible by 2).

Divisibility by 5. Number is divisible by 5 if and only if its last digit is either 0 or 5.

For example, 25 is divisible by 5 (last digit is 5), while 43 is not divisible by 5 (last digit is neither 0 nor 5).

Divisibility by 10. Number is divisible by 10 if and only if its last digit is 0.

For example, 260 is divisible by 10 (last digit is 0), while 57 is not divisible by 10 (last digit is not 0).

Divisibility by 4. Number that has at least 3 digits is divisible by 4 if and only if two-digit number that is formed from last two digits of number is divisible by 4.

For example, 324 is divisible by 4 (24 is divisible by 4), while 4318 is not divisible by 4 (18 is not divisible by 4).

Divisibility by 25. Number that has at least 3 digits is divisible by 25 if and only if two-digit number that is formed from last two digits of number is divisible by 25.

For example, 325 is divisible by 25 (25 is divisible by 25), while 4351 is not divisible by 25 (51 is not divisible by 25).

Divisibility by 3. Number is divisible by 3 if and only if sum of its digits is divisible by 3.

For example, 207 is divisible by 3 (sum of digits 2+0+7=9 is divisible by 3), while 5423 is not divisible by 3 (sum of digits 5+4+2+3=14 is not divisible by 3).

Divisibility by 9. Number is divisible by 9 if and only if sum of its digits is divisible by 9.

For example, 207 is divisible by 9 (sum of digits 2+0+7=9 is divisble by 9), while 543 is not divisible by 9 (sum of digits 5+4+3=12 is not divisible by 9).

Now, solve following problems.

Exercise 1. Determine whether 45867 is divisible by 2.

45867 is not divisible by 2 because last digit (7) is not divisible by 2.

Next exercise.

Exercise 2. Determine whether 45867 is divisible by 3.

Find sum of digits: 4+5+8+6+7=30.

Since 30 is divisible by 3 then 45867 is divisible by 3.

Next exercise.

Exercise 3. Determine whether 45867 is divisible by 4.

Since 67 is not divisible by 4 then 45867 is not divisible by 4.

Next exercise.

Exercise 4. Determine whether 45867 is divisible by 5.

Since last digit (7) is neither 0 nor 5 then 45867 is not divisible by 5.

Next exercise.

Exercise 5. Determine whether 45867 is divisible by 9.

Find sum of digits: 4+5+8+6+7=30.

Since 30 is not divisible by 9 then 45867 is not divisible by 9.

Next exercise.

Exercise 6. Determine whether 45860 is divisible by 10.

Since last digit is 0 then 45860 is divisible by 10.

Next exercise.

Exercise 7. Determine whether 45875 is divisible by 25.

Since 75 is divisible by 25 then 45875 is divisible by 25.

Next exercise.

Exercise 8. Determine whether ${107}\cdot{96}$ is divisible by 3.

107 is not divisible by 3, because sum of digits is not divisible by 3, but 96 is divisible by 3.

Thus, ${107}\cdot{96}$ is divisible by 3.

Last exercise.

Exercise 9. Determine whether ${150}\cdot{918}$ is divisible by 225.

Since 150 is divisible by 25 (because 50 is divisible by 25) and 918 is divisible by 9 (sum of digits 9+1+8=18 is divisible by 9) then ${150}\cdot{918}$ is divisible by ${25}\cdot{9}={225}$.