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# Divisibility

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### Divisibility

1. 1. Before we study divisibility, we mustremember the division algorithm. rdividend = (divisor ⋅ quotient) + remainder
2. 2. Anumber is divisible by anothernumber if the remainder is 0 andquotient is a natural number.
3. 3.  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.
4. 4.  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.Example: Check if each number is divisibleby 2.a. 108 b. 466 c. 87 682 d. 68241e. 76 543 010
5. 5.  Divisibility by 3: A natural number is divisible by 3 if the sum of the digits in the number is multiple of 3.Example: Determine whether thefollowing numbers are divisible by 3 ornot.a) 7605b) 42 145c) 555 555 555 555 555
6. 6.  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.Example: Determine whether thefollowing numbers are divisible by 4 ornot.a) 7600b) 47 116c) 985674362549093
7. 7. Example: 5m3 is a three-digit numberwhere m is a digit. If 5m3 is divisibleby 3, find all the possible values of m.Example: a381b is a five-digit numberwhere a and b are digits. If a381b isdivisible by 3, find the possible valuesof a + b.
8. 8. Example: t is a digit. Find all the possiblevalues of t if:a) 187t6 is divisible by 4.b) 2741t is divisible by 4.
9. 9. Divisibility by 5:A natural number is divisible by 5 if itslast digit is 0 or 5.Example: m235m is a five-digit numberwhere m is a digit. If m235m is divisibleby 5, find all the possible values of m.
10. 10. Divisibility by 6:A natural number is divisible by 6 if it isdivisible by both 2 and 3.Example: Determine whether thefollowing numbers are divisible by 6 ornot.a) 4608b) 6 9030c) 22222222222
11. 11. Example: 235mn is a five-digit numberwhere m and n are digits. If 235mn isdivisible by 5 and 6, find all the possiblepairs of m, n.
12. 12. Divisibility by 8:A natural number is divisible by 8 if thenumber formed by last three digits isdivisible by 8.Example: Determine whether thefollowing number is divisible by 8 ornot.a) 5 793 128b) 7265384c) 456556
13. 13. Divisibility by 9:A natural number is divisible by 9 if thesum of the digits of the number isdivisible by 9.Example: 365m72 is a six-digit numberwhere m is a digit. If 365m72 isdivisible by 9, find all the possiblevalues of m.Example: 5m432n is a six-digit numberwhere m and n are digits. If 5m432n isdivisible by 9, find all the possiblevalues of m + n.
14. 14. Divisibility by 10:A natural number is divisible by 10 if itsunits (last) digit is 0.Example: is 3700 divisible by 10?
15. 15. Divisibility by 11:A natural number is divisible by 11 if thedifference 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 by11?