The Division Algorithm

Before we study divisibility, we must remember the division algorithm.

€

b ac)

r

dividend = (divisor ⋅ quotient) + remainder

A number is divisible by another number if the remainder is 0 and quotient is a natural number.

Divisibility

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:

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 241e. 76 543 010

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.a) 7605b) 42 145c) 555 555 555 555 555

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.a) 7600b) 47 116c) 985674362549093

Divisibility Rules

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.

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.

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.

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.

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.a) 4608b) 6 9030c) 22222222222

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.

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.a) 5 793 128b) 7265384c) 456556

Divisibility RulesDivisibility 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.

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?

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?