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 the divisors 2 through 11. Specifically, it explains that a number is divisible by: - 2 if the last digit is even - 3 if the sum of the digits is divisible by 3 - 4 if the last two digits are divisible by 4 - 5 if the last digit is 0 or 5 - 6 if it is divisible by both 2 and 3 - 8 if the last three digits are divisible by 8 - 9 if the sum of the digits is divisible by 9 - 10 if the last digit is 0 - 11 if the digit

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?