The document discusses divisibility rules for natural numbers. It begins by introducing the division algorithm and defining divisibility. It then provides divisibility rules and examples for numbers being divisible by 2, 3, 4, 5, 6, 8, 9, 10, and 11. Specific rules include a number being divisible by 2 if the last digit is even, divisible by 3 if the sum of the digits is divisible by 3, and divisible by 4 if the last two digits are a multiple of 4.

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?