# Divisibility Rules For numbers 10 – 19

###### Divisibility Rules For numbers 10 – 19

The rules for numbers greater than 10 especially numbers between 10 and 20 work in this way – Given a number y = 10T + U, where U is the unit digit number and T is the number formed by the rest digits.

A number y is divisible by N (where N is a number greater than 10) if (N-10)T – U is divisible by N. That is, find the difference between the number N and 10

Note: This is true of all numbers greater than 10 but is not efficient in some cases

The rule for 11 using the above:

A number y is divisible by 11 if and only if (11 – 10)T – U is divisible by 11

The difference between 11 and 10 is 1

Example 1: Test whether 132 is divisible by 11,

Multiply 13 by 1 and subtract 2 from the product. That is (1×13) – 2 = 13 -2 = 11,  a number divisible by 11.

Example 2: Test whether 1023 is divisible by 11,

Multiply 102 by 1 and subtract 3 from the product. That is (1×102) – 3 =  102 – 3 = 99 a number divisible by 11.

The rule for 12

Old: A number is divisible by 12 if and only if the number is divisible by 3 and 4.

Using the above: A number y is divisible by 12 if and only if (12 -10)T – U (that is 2T – U ) is divisible by 12

The difference between 12 and 10 is 2

Example 1: Test whether 156 is divisible by 12,

Multiply 15 by 2 and subtract 6 from the product. That is (2 x 15) – 6 = 30 – 6 = 24, gives a number divisible by 12.

Example 2: Test whether 576 is divisible by 12,

Multiply 57 by 2 and subtract 6 from the product. That is (2 x 57) – 6 = 114 – 6 = 108, a number divisible by 12.

The rule for 13

Using the above: A number y is divisible by 13 if and only if (13 -10)T – U (that is 3T – U ) is divisible by 13.

The difference between 13 and 10 is 3

Example 1: Test whether 169 is divisible by 13,

Multiply 16 by 3 and subtract 9 from the product. That is (3 x 16) – 9 = 48 – 9 = 39, a number divisible by 13.

Example 2: Test whether 312 is divisible by 13,

Multiply 31 by 3 and subtract 2 from the product. That is (3 x 31 ) – 2 = 93 – 2 = 91, a number divisible by 13

###### Rules for 14

(a) A number is divisible by 14 if it is divisible by 2 and 7

(b)  A number y is divisible by 14 if and only if (14 -10)T – U (that is 4T – U ) is divisible by 14.

(c)  A number is divisible by 14 if 6T + 2U is divisible by 14

Using rule (b)

The difference between 14 and 10 is 4

Example 1: Test whether 98 is divisible by 14,

Multiply 9 by 4 and subtract 8 from the product. That is (4 x 9) – 8 = 36 – 8 = 28, a number divisible by 14.

Using rule (c)

Example : Test whether 182 is divisible by 14,

182: (6 x 18 ) +( 2 x 2) = 108 + 4 = 112,

112: (6 x 11) + 2 x 2 = 70 = 14 x 5,  a number divisible by 14.

The rule for 15

(a): A number is divisible by 15 if and only if the number is divisible by 3 and 5.

(b) : A number y is divisible by 15 if and only if (15 -10)T – U (that is 5T – U ) is divisible by 15.

(c) : A number y is divisible by 15 if and only if  5T + 2U ) is divisible by 15.

Using rule (b)

The difference between 15 and 10 is 5

Example 1: Test whether 135 is divisible by 15,

135:  (5 x 13) – 5 = 65 – 5 = 60 = 15 x 4, a number divisible by 15.

Using rule (c)

Example : Test whether 225 is divisible by 15,

225:  is (5 x 22 )  + (2 x 5) = 110 + 10 = 120 = 15 x 8, a number divisible by 15.

The rule for 16

(a) A number y is divisible by 16 if and only if (16 -10)T – U (that is 6T – U ) is divisible by 16.

(b) A number is divisible by 16 if 4T + 2U is divisible by

The difference between 16 and 10 is 6

Example 1: Test whether 128 is divisible by 16,

Multiply 12 by 6 and subtract 8 from the product. That is (12 x 6) – 8 = 72 – 8 = 64, a number divisible by 16.

Example 2: Test whether 256 is divisible by 16,

Multiply 25 by 6 and subtract 6 from the product. That is (6 x 25 ) –  6 = 150 – 6 = 144, a number divisible by 16.

The rule for 17

(a): A number y is divisible by 17 if and only if (17 -10)T – U (that is 7T – U ) is divisible by 17.

(b) A number is divisible by 17 if 3T + 2U is divisible by 17

Using rule (a)

The difference between 17 and 10 is 7

Example : Test whether 68 is divisible by 17,

68:  (7 x 6) – 8 = 42 – 8 = 34 = 17 x 2, a number divisible by 17.

Using rule (b)

Example 2: Test whether 204 is divisible by 17,

204: (7 x 20) – 4 = 140 – 4 = 136, a number divisible by 17.

If one is not sure that 136 is divisible by 17, this number can be tested recursively. That is, (7 x 13) – 6 = 91 – 6 = 85= 17 x 5, a number divisible by 17.

The rule for 18

(a): A number is divisible by 18 if and only if the number is even and divisible by 9.

(b): A number y is divisible by 18 if and only if (18 -10)T – U (that is 8T – U ) is divisible by 18.

(c): A number is divisible by 18 if 2T + 2U (that is, 2(T + U) is divisible by 18

Using rule (b)

The difference between 18 and 10 is 8

Example : Test whether 72 is divisible by 18,

72:  (8 x 7) – 2 = 56 – 2 = 54 = 18 x 3, a number divisible by 18.

Using rule (c) that is testing for 2 and (T + U)

Example : Test whether 126 is divisible by 18,

Since 126 is even and 126: 12 + 6 = 18, 126 is divisible by 18.

The rule for 19

Using the above: A number y is divisible by 19 if and only if (19 -10)T – U (that is 9T – U ) is divisible by 19.

The difference between 19 and 10 is 9

Example 1: Test whether 76 is divisible by 19,

Multiply 7 by 9 and subtract 6 from the product. That is (9 x 7) – 6 = 63 – 6 = 57, a number divisible by 19.

Example 2: Test whether 209 is divisible by 19,

Multiply 20 by 9 and subtract 9 from the product. That is (9 x 20) – 9 = 180 – 9 = 171, a number divisible by 19.

If one is not sure that 171 is divisible by 19, this number can be tested recursively. That is, (9 x 17) – 1 = 153 – 1 = 152, which is a number divisible by 19.

A simpler Rule for 19:

A number y is divisible by 19 if and only if T + 2U is divisible by 19.

Example 1: Test whether 209 is divisible by 19,

Using T + 2U, 20 + (2 x 9) = 38, a number divisible by 19.

Example 2: Test whether 437 is divisible by 19,

Using T + 2U, 43 + (2 x 7) = 43 + 14 = 57, a number divisible by 19.

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