CAT Divisibility rules of 2, 3, 4, 5, 6, 7, 8, 9, 11 and 13. - Practice Questions & MCQ

Introduction

The concept of divisibility is essential in quantitative aptitude as it helps in simplifying calculations and solving problems efficiently. Understanding the divisibility rules of various numbers can significantly save time during exams, especially management entrance exams. In this study note, we will learn the divisibility rules of numbers 2, 3, 4, 5, 6, 7, 8, 9, 11, and 13, along with some examples.

Divisibility Rule of 2

A number is divisible by 2 if its unit digit is an even number (0, 2, 4, 6, or 8).

Example:

• 7546 is divisible by 2 because the unit digit is 6.
• 9243 is not divisible by 2 because the unit digit is 3.

Divisibility Rule of 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

Example:

• 372 is divisible by 3 because 3 + 7 + 2 = 12, and 12 is divisible by 3.
• 815 is not divisible by 3 because 8 + 1 + 5 = 14, and 14 is not divisible by 3.

Divisibility Rule of 4

A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

Example:

• 864 is divisible by 4 because 64 is divisible by 4.
• 936 is not divisible by 4 because 36 is not divisible by 4.

Divisibility Rule of 5

A number is divisible by 5 if its unit digit is either 0 or 5.

Example:

• 4250 is divisible by 5 because the unit digit is 0.
• 731 is not divisible by 5 because the unit digit is 1.

Divisibility Rule of 6

A number is divisible by 6 if it is divisible by both 2 and 3.

Example:

• 582 is divisible by 6 because it is divisible by 2 and 3.
• 139 is not divisible by 6 because it is not divisible by either 2 or 3.

Divisibility Rule of 7

A number is divisible by 7 if the difference between twice the unit digit and the number formed by the remaining digits is divisible by 7.

Example:

• 399 is divisible by 7 because (2 x 9) - 39 = 18 - 39 = -21, which is divisible by 7.
• 685 is not divisible by 7 because (2 x 5) - 68 = 10 - 68 = -58, which is not divisible by 7.

Divisibility Rule of 8

A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

Example:

• 1696 is divisible by 8 because 696 is divisible by 8.
• 8125 is not divisible by 8 because 125 is not divisible by 8.

Divisibility Rule of 9

A number is divisible by 9 if the sum of its digits is divisible by 9.

Example:

• 567 is divisible by 9 because 5 + 6 + 7 = 18, and 18 is divisible by 9.
• 954 is not divisible by 9 because 9 + 5 + 4 = 18, and 18 is not divisible by 9.

Divisibility Rule of 11

A number is divisible by 11 if the difference between the sum of digits at odd places (1st, 3rd, 5th, etc.) and the sum of digits at even places (2nd, 4th, 6th, etc.) is either 0 or divisible by 11.

Example:

• 1243 is divisible by 11 because (1 + 4) - (2 + 3) = 5 - 5 = 0, which is divisible by 11.
• 8531 is not divisible by 11 because (8 + 3) - (5 + 1) = 11 - 6 = 5, which is not divisible by 11.

Divisibility Rule of 13

Here's the step-by-step procedure:

1. Take the number you're checking for divisibility by 13. Let's call this number "N".
2. Multiply the last digit of "N" by 9 and subtract this from the rest of the number.
3. If the result is 0 or if it is a multiple of 13, then the original number is divisible by 13.
4. If the result is neither 0 nor a multiple of 13, repeat the process on this new number.

Example:

Check the divisibility of 507. Multiply the last digit (7) by 9 to get 63. Subtract 63 from the rest of the number (50), giving -13. -13 is a multiple of 13 (since -13 = -1*13), so 507 is divisible by 13.

Tips and Tricks

• Practise mental calculations to quickly identify divisibility.
• Memorise the divisibility rules for the digits 2, 3, 4, 5, 6, 7, 8, 9, 11, and 13.
• Divide a number into smaller parts to perform divisibility checks.
• When the last digit is 0 or 5, the number is divisible by 5.
• When the sum of digits is divisible by 3, the number is divisible by 3.