Divisibility

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Divisibility - 3, 9, 11

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Module 2 of 8

Overview

2.3 Divisor is $3$ or $9$

Rule: Remainder when a number is divided by

3

9

is the remainder of the sum of digits of the number divided by 3 or 9.

Explanation

10 = 9 + 1

9

is divisible by

3

and

9

\text{Rem} \left(\dfrac{10^{n}}{3} \right) = \text{Rem} \left(\dfrac{(9 + 1)^{n}}{3} \right) = \text{Rem} \left(\dfrac{1^n}{3} \right) = 1

\text{Rem} \left(\dfrac{10^{n}}{9} \right)= \text{Rem} \left(\dfrac{(9 + 1)}{9} \right) = \text{Rem} \left(\dfrac{1^n}{9} \right) = 1

n \ge 0

, when

10^n

is divided by

3

9

, the remainder is always

1

. The following example shows the extension of this concept for the sum of digits rule.

Example 4

What is the remainder when

34585

is divided by

3

Solution

34585 = (3 \times 10^4) + (4 \times 10^3) + (5 \times 10^2) + (8 \times 10^1) + (5 \times 10^0)

Let the remainder be

x

x = \text{Rem} \left(\dfrac{34585}{3} \right)

= \text{Rem} \left(\dfrac{(3 \times 10^4) + (4 \times 10^3) + (5 \times 10^2)+ (8 \times 10^1) + (5 \times 10^0)}{3} \right)

\text{Rem} \left(\dfrac{10^n}{3} \right) = 1

x = \text{Rem} \left(\dfrac{(3 \times 1) + (4 \times 1) (5 \times 1) (8 \times 1) (5 \times 1)}{3}\right)

= \text{Rem} \left(\dfrac{3 + 4 + 5 + 8 + 5}{3} \right) = \text{Rem} \left(\dfrac{25}{3} \right)= \text{Rem} \left(\dfrac{2 + 5}{3} \right)

= \text{Rem}\left(\dfrac{7}{3} \right)= 1

Answer:

1

The above example was to explain the rationale for the sum of digits rule. Going forward, successive digit sums. should be used to obtain the remainder. The following will help to improve your speed.

1) Keep adding the digits till you're left with a single digit number.
2) Remove all

9

s from the digits or sum of digits.

Example 5

What is the remainder when

45678349

is divided by

9

Solution

We can add

2

digits at a time and then add the digits of the results. This is a relatively faster way to find the remainder.

(4 + 5) + (6 + 7) + (8 + 3) + 4 + (9)

= 9 + 13 + 11 + 4 + 9

9

's can be removed and rest of the digits can be added again.

1 + 3 + 1 + 1 + 4 = 10

Once again we apply digit sum.

1 + 0 = 1

Answer:

1

2.4 Divisor is $11$

Rule: Remainder when a number is divided by

11

equals the remainder when the sum of alternating digits starting from the units' place is subtracted from the sum of the remaining digits and then divided by

11

.

Explanation

\text{Rem} \left(\dfrac{10^0}{11} \right) = 1 ; \text{Rem} \left(\dfrac{10^1}{11} \right) = 10 ; \text{Rem} \left(\dfrac{10^2}{11} \right) = 1 ; \text{Rem} \left(\dfrac{10^3}{11} \right) = 10

As noted above,

1

and

10

are the remainders when even and odd powers of

10

are divided by

11

respectively.

As any number of divisors can be subtracted or added to a remainder, remainder for odd powers of

10

can be written as

10 - 11 = -1

∴

\text{Rem} \left(\dfrac{10^n}{11} \right) = 1

n

is even and -1 if

n

is odd.

Example 6

What is the remainder when

94265

is divided by

11

Solution

94265 = (9 \times 10^{4}) + (4 \times 10^{3}) + (2 \times 10^{2}) + (6 \times 10^{1}) + (5 \times 10^{0})

Let the remainder be

x

x = \text{Rem} \left(\dfrac{94265}{11} \right)

= \text{Rem} \left(\dfrac{(9 \times 10^4) + (4 \times 10^3) + (2 \times 10^2) + (6 \times 10^1) + (5 \times 10^0)}{11} \right)

\text{Rem} \left(\dfrac{10^n}{11} \right) = 1

n

is even and -1 if

n

is odd. ,

x = \text{Rem} \left(\dfrac{(9 \times 10^4) + (4 \times -1) + (2 \times 1) + (6 \times -1) + (5 \times 1)}{3} \right)

= \text{Rem} \left(\dfrac{9 - 4 + 2 - 6 + 5}{11} \right)= \text{Rem} \left(\dfrac{6}{11} \right) = 6

Alternatively

The rule can directly be applied. Remainder when

94265

is divided by

11

= (5 + 2 + 9) - (6 + 4) = 6

Answer:

6

CAT 2025 Lesson : Divisibility - Divisibility - 3, 9, 11

2.3 Divisor is 333 or 999

Example 4

Solution

Example 5

Solution

2.4 Divisor is 111111

Example 6

Solution

2.3 Divisor is $3$ or $9$

2.4 Divisor is $11$