1. DIVISIBILITY RULE EXTENSION
DivisibilityRule forPrime Numbers
DivisibilityRule for7
Is 623 divisibleby7?
*Multiply the lastdigit of 623 by 2 then SUBTRACTthe remaining digitsi.e. 62 to the product.
62 – 2(3) = 56
Since 56 isdivisibleby7,therefore 623 is divisibleby7.
DivisibilityRule for13
Is 16185 divisibleby13?
*Multiply the lastdigit of 16185 by 4 and ADDthe remaining digits i.e. 1618 to the product.
1618 + 4(5) = 1638
163 + 4(8) = 195
19 + 4(5) = 39
Since 39 isdivisibleby13, therefore 16185 is divisible by13.
DivisibilityRule for31
Is 24459 divisibleby31?
*Multiply the lastdigit of 24459 by 3 and SUBTRACT theremaining digits i.e. 2445 to theproduct.
2445 – 3(9) = 2418
241 – 3(8) = 217
21 – 3(7) = 0
Since 0 is divisible byanynumbertherefore itisdivisible by31,thus 24459 is alsodivisible by31.
The numberthat multiplytothe lastdigitiscalledDIVISIBILITYFACTORforprime numbers. Inour
example for7 we have 2, for 13 we have 4, and for 31 we have 3.
Typesof DIVISIBILITYFACTOR
A. Positive DivisibilityFactor
In Positive DivisibilityFactormultiplythe DivisibilityFactorto the lastdigitof the numberthenADD to
the remainingdigits.
2. B. Negative DivisibilityFactor
In Negative DivisibilityFactormultiplythe DivisibilityFactortothe lastdigitof the numberthen
SUBTRACT to the remainingdigits.
How to findPositiveDivisibilityFactor?
For numbersending in 9, the Positive Divisibility Factor is onemore than the remaining digits.
e.g. 9 DivisibilityFactor= 0 + 1 = 1
19 DivisibilityFactor= 1 + 1 = 2
29 DivisibilityFactor= 2 + 1 = 3
59 DivisibilityFactor= 5 + 1 = 6
For numberending in 3, multiply the numberby 3 to make the lastdigit equal to 9, then apply the rule
fornumbersending in 9.
e.g. 13 13(3) = 39 DivisibilityFactor= 3 + 1 = 4
23 23(3) = 69 DivisibilityFactor= 6 + 1 = 7
43 43(3) = 129 DivisibilityFactor= 12 + 1 = 13
For numbersending in 7 and 1 multiply the numbersby 7 or 9 so thatthe lastdigit becomes9 then apply
the rule fornumbersending in 9.
On the otherhand,for prime numbersendingin7and 1 it isbetterto use the Negative Divisibility
Factor to make it easier.
How to findNegativeDivisibilityFactor?
For numbersending in 1, justsimply drop the last digit and thenumberthatis left is the Divisibility
Factor.
e.q 11 DivisibilityFactor= 1
31 DivisibilityFactor= 3
41 DivisibilityFactor= 4
71 DivisibilityFactor= 7
For numbersending in 7, multiply the numberby 3 to make thelast digit equal to 1 then apply the rule
fornumbersending with1.
3. e.g. 7 7(3) = 21 DivisibilityFactor= 2
17 17(3) = 51 DivisibilityFactor= 5
37 37(3) = 111 DivisibilityFactor= 11
Summary:
For numbersending with1 and 7, useNegativeDivisibility Factorand fornumbersending in 3 and 9 use
PositiveDivisibility Factor.
Take note: ThisDivisibilityFactorscanbe used even ifthe number isnot a prime.
Is 9597 divisible by21?
Use Negative DivisibilityFactor
Use the rule fornumbersendingin1.
21 DivisibilityFactor2
959 – 2(7) = 945
94 – 2(5) = 84
since 84 isdivisibleby21, therefore 9597 isalso divisible by21.
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