 maths > wholedivisors

Simplification of Digits for Divisibility Tests

what you'll learn...

overview

The following are explained.

•  Reducing the number of digits of the dividend to simplify divisibility test is explained.

•  Divisibility test for 7$7$

•  Divisibility test for 13$13$

Divisibility: Simplification in Digits

The "Divisibility Test: Simplification by Subtraction" is "difference between the dividend and a multiple of divisor is checked for divisibility by divisor"

The "Divisibility Test: Simplification by Division" is
• dividend can be divided by a co-prime (of divisor) factor and result is checked for divisibility
• dividend and divisor are divided by common factors and the result is checked for divisibility

The "simplification by subtraction" and "simplification by division" are further refined.

Take a number 2352$2352$ and check for divisibility by 7$7$.
We can simplify the number as

divisibility of 2352$2352$

subtracting a multiple of 7$7$
divisibility $\to 2352 - 7 \times 3 \times 2$

divisibility $\to 2352 - 42 = 2310$

$2$ and $5$ are co-primes of divisor $7$
divisibility $\to \frac{2310}{10} = 231$
repeating subtraction of a multiple of $7$
divisibility $\to 231 - 7 \times 3 \times 1 = 210$

repeating $2$ and $5$ co-prime simplification
divisibility $\to \frac{210}{10} = 21$

$21$ is divisible by $7$, and so $2352$ is divisible by $7$.

Note: The multiple of $7$ is chosen with two factors

•  First $3$ is chosen because, $7 \times 3 = 21$ where the units place is $1$

•  Second $2$ is chosen because the units digit of the number is $2$ and on subtraction that makes the units digit $0$. That is, $2352 - 21 \cdot 2 = 2310$. This facilitates the co-prime of $10$ step, thereby reducing the number of digits.

Another example of combining "simplification by subtraction" and "simplification by division".

Take a number $2352$ and check for divisibility by $13$.

Look at the multiples of $13$ : $26$, $39$, etc.
The multiple $39$ is interesting as $39 \cdot n + n$ is a multiple of $10$.

We can simplify the number $2352$ as

divisibility of $2352$

adding a multiple of $13$
divisibility $\to 2352 + 13 \times 3 \times 2$

divisibility $\to 2352 + 78 = 2430$

$2$ and $5$ are co-primes of divisor $13$
divisibility $\to \frac{2430}{10} = 243$
repeating addition of a multiple of $13$
divisibility $\to 231 + 13 \times 3 \times 1 = 270$

repeating $2$ and $5$ co-prime simplification
divisibility $\to \frac{270}{10} = 27$

$27$ is not divisible by $13$, and so $2352$ is not divisible by $13$.

Note: The multiple of $13$ is chosen with two factors

•  First $3$ is chosen because, $13 \times 3 = 39$ where the units place is $9$

•  Second $2$ is chosen because the units digit of the number is $2$ and on addition that makes the units digit $0$. That is, $2352 + 39 \cdot 2 = 2430$. This facilitates the co-prime of $10$ step, thereby reducing the number of digits.

Simplification in Digits : Consider the multiples of the divisor and choose the one that ends $1$ or $9$. By using Simplification by Subtraction and Simplification by Division, reduce the number of digits of the dividend. The reduced dividend can be checked for divisibility, and the simplification can be used repeatedly.

Divisibility by $7$

The divisibility test of $7$ is made easiler by "simplification in digits as $7 \times 3 = 21$".

If the dividend is $10 A + B$, (where $B$ is a single digit number)

then the number can be modified to $10 A + B - 21 B$ (simplification by subtraction)

which equals $10 A - 20 B = 10 \times \left(A - 2 B\right)$.

Since $7$ is co-prime to both $2$ and $5$, the divisibility test is done on $A - 2 B$ (simplification by division). This, in effect, reduced the number of digits in $10 A + B$.

Note: $A - 2 B$ can further be simplified using the same procedure.

Test for Divisibility by $7$ : Reduce the number of digits by iteration.

Remove the units digit from the number to get a modified number. Subtract double of the removed digit in units place from the modified number.

Perform the divisibility test on the difference.

Is $2008$ divisible by $7$?
Following the process
$2008$
$\to 200 - 2 \times 8$
$\to 184$
$\to 18 - 2 \times 4$
$\to 10$

$10$ is not divisible by $7$, and so it is concluded that the number is not divisible by $7$.

Is $406$ divisible by $7$?
Following the process
$406$
$\to 40 - 2 \times 6$
$\to 28$

$28$ is divisible by $7$, and so it is concluded that the number is divisible by $7$.

Divisibility by $13$

The divisibility test of $13$ is made easiler by "simplification in digits as $13 \times 3 = 39$".

If the given dividend is $10 A + B$, (where $B$ is a single digit number)

then the dividend can be modified to $10 A + B + 39 B$ (simplification by addition)

which equals $10 A + 40 B = 10 \left(A + 4 B\right)$.

Since $13$ is co-prime to both $2$ and $5$, the divisibility test is done on $A + 4 B$ (simplification by division). This, in effect, reduced the number of digits in $10 A + B$.

Note: $A + 4 B$ can further be simplified using the same procedure.

Test for Divisibility by $13$ : Reduce the number of digits by iteration.

Remove the units digit from the dividend to get a modified number. Add $4$ times of the removed units digit to the modified number.

Perform the divisibility test on the sum.

Is $2008$ divisible by $13$?
Following the process
$2008$
$\to 200 + 4 \times 8$
$\to 232$
$\to 23 + 4 \times 2$
$\to 31$

$31$ is not divisible by $13$, and so it is concluded that the number is not divisible by $13$.

Is $416$ divisible by $13$?
Following the process
$416$
$\to 41 + 4 \times 6$
$\to 65$
$\to 6 + 4 \times 5$
$\to 26$

$26$ is divisible by $13$, and so it is concluded that the number is divisible by $13$.

Divisibility by $14$

Consider the number $2 \times 7 \times 2351$. It is divisible by $14$ as it has $2$ and $7$ as factors.

Test for Divisibility by $14$ : If a dividend is divisible by $2$ and divisible by $7$ then it is divisible by $14$

Is $2008$ divisible by $14$?
The number is divisible by $2$ but not by $7$. So it is not divisible by $14$.

Is $924$ divisible by $14$?
The number is divisible by $2$ and also divisible by $7$. So it is divisible by $14$.

summary

Simplification in Digits : Consider the multiples of the divisor and choose the one that ends $1$ or $9$. By using Simplification by Subtraction and Simplification by Division, reduce the number of digits of the dividend. The reduced dividend can be checked for divisibility, and the simplification can be used repeatedly.

Test for Divisibility by $7$ : Reduce the number of digits by iteration.

Remove the units digit from the number to get a modified number. Subtract double of the removed digit in units place from the modified number.

Perform the divisibility test on the difference.

Test for Divisibility by $13$ : Reduce the number of digits by iteration.

Remove the units digit from the dividend to get a modified number. Add $4$ times of the removed units digit to the modified number.

Perform the divisibility test on the sum.

Test for Divisibility by $14$ : If a dividend is divisible by $2$ and divisible by $7$ then it is divisible by $14$

Outline

The outline of material to learn "Divisibility in Whole Numbers" is as follows.

→   Classification as odd, even, prime, and composite

→   Factors, Multiples, Prime factorization

→   Highest Common Factor

→   Lowest Common Multiple

→   Introduction to divisibility tests

→   Simple Divisibility Tests

→   Simplification of Divisibility Tests

→   Simplification in Digits for Divisibility Tests