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Divisibility

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Divisibility

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