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# Divisibility

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• 1. The Division Algorithm
• 2. Before we study divisibility, we must remember the division algorithm.
r
dividend = (divisor ⋅ quotient) + remainder
• 3. A number is divisible by another number if the remainder is 0 and quotient is a natural number.
Divisibility
• 4. If a number is divided by itself then quotient is 1.
If a number is divided by 1 then quotient is itself.
If 0 is divided by any none zero number then quotient is 0.
If any number is divided by zero then quotient is undefined.
Some remarks:
• 5. Divisibility by 2:
A natural number is divisible by 2 if it is even, i.e. if its units (last) digit is 0, 2, 4, 6, or 8.
Divisibility Rules
Example: Check if each number is divisible by 2.
a. 108 b. 466 c. 87 682 d. 68 241
e. 76 543 010
• 6. Divisibility by 3:
A natural number is divisible by 3 if the sum of the digits in the number is multiple of 3.
Divisibility Rules
Example: Determine whether the following numbers are divisible by 3 or not.
7605
42 145
c) 555 555 555 555 555
• 7. Divisibility by 4:
A natural number is divisible by 4 if the last two digits of the number are 00 or a multiple of 4.
Divisibility Rules
Example: Determine whether the following numbers are divisible by 4 or not.
7600
47 116
c) 985674362549093
• 8. Divisibility Rules
Example: 5m3 is a three-digit number where m is a digit. If 5m3 is divisible by 3, find all the possible values of m.
Example: a381b is a five-digit number where a and b are digits. If a381b is divisible by 3, find the possible values of a + b.
• 9. Divisibility Rules
Example: t is a digit. Find all the possible values of t if:
a) 187t6 is divisible by 4.
b) 2741t is divisible by 4.
• 10. Divisibility Rules
Divisibility by 5:
A natural number is divisible by 5 if its last digit is 0 or 5.
Example: m235m is a five-digit number where m is a digit. If m235m is divisible by 5, find all the possible values of m.
• 11. Divisibility Rules
Divisibility by 6:
A natural number is divisible by 6 if it is divisible by both 2 and 3.
Example: Determine whether the following numbers are divisible by 6 or not.
4608
6 9030
c) 22222222222
• 12. Divisibility Rules
Example: 235mn is a five-digit number where m and n are digits. If 235mn is divisible by 5 and 6, find all the possible pairs of m, n.
• 13. Divisibility Rules
Divisibility by 8:
A natural number is divisible by 8 if the number formed by last three digits is divisible by 8.
Example: Determine whether the following number is divisible by 8 or not.
5 793 128
7265384
456556
• 14. Divisibility Rules
Divisibility by 9:
A natural number is divisible by 9 if the sum of the digits of the number is divisible by 9.
Example: 365m72 is a six-digit number where m is a digit. If 365m72 is divisible by 9, find all the possible values of m.
Example: 5m432n is a six-digit number where m and n are digits. If 5m432n is divisible by 9, find all the possible values of m + n.
• 15. Divisibility Rules
Divisibility by 10:
A natural number is divisible by 10 if its units (last) digit is 0.
Example: is 3700 divisible by 10?
• 16. Divisibility Rules
Divisibility by 11:
A natural number is divisible by 11 if the difference between the sum of the odd-numbered digits and the sum of the even-numbered digits is a multiple of 11.
Example: is 5 764 359 106 divisible by 11?