Divisibility Rules and Remainders | CSE Reviewer

Numbers are part of daily life, and divisibility helps you quickly know if one number can be divided by another without any leftover. This lesson is important because it makes math faster and easier, and it also helps you answer school exercises and Civil Service Exam questions that use divisibility and remainders.

Divisibility

Divisibility means that one number can be divided by another number evenly. When you divide and there is no remainder, the number is divisible. For example, \(12\) is divisible by \(3\) because \(12 \div 3 = 4\) with no remainder.

This idea is useful because it helps you check answers without doing long division every time. It also helps you find factors and multiples more quickly, which is a common skill in school math.

Example: \(20\) is divisible by \(5\) because \(20 \div 5 = 4\). There is no leftover, so the division is exact.

Meaning: Divisibility means a number can be split equally. If nothing is left after dividing, then it is divisible.

Divisibility Rule for 2

A number is divisible by \(2\) if its last digit is \(0\), \(2\), \(4\), \(6\), or \(8\). These are the even numbers. If the last digit is odd, the number is not divisible by \(2\).

This rule is easy to use because you only look at the last digit. You do not need to divide the whole number. This saves time, especially in tests and class exercises.

Tip: If the last digit is even, the number is divisible by \(2\). This is the fastest rule to check.

Example: \(148\) is divisible by \(2\) because it ends in \(8\). But \(147\) is not divisible by \(2\) because it ends in \(7\).

Meaning: A number is divisible by \(2\) when it ends in an even digit. Just check the last digit and you will know fast.

Divisibility Rule for 5

A number is divisible by \(5\) if its last digit is \(0\) or \(5\). This rule is very easy because only two digits matter. Numbers like \(10\), \(25\), \(40\), and \(85\) follow this rule.

You can use this when checking prices, counting objects, or solving exam items. Many learners remember this rule because it is simple and quick.

Tip: Numbers ending in \(0\) are always divisible by \(10\), so they are also divisible by \(5\).

Example: \(135\) is divisible by \(5\) because it ends in \(5\). But \(132\) is not divisible by \(5\) because it ends in \(2\).

Meaning: If a number ends in \(0\) or \(5\), it is divisible by \(5\). That is the whole rule.

Divisibility Rule for 10

A number is divisible by \(10\) if its last digit is \(0\). This is the easiest rule of all. Any number ending in \(0\) can be divided by \(10\) exactly.

This rule is useful when dealing with money, counting by tens, and checking whole tens. It is also often used together with the rule for \(5\) because every number divisible by \(10\) is also divisible by \(5\).

Tip: If a number ends in \(0\), you do not need to check further. It is divisible by \(10\) right away.

Example: \(230\) is divisible by \(10\) because it ends in \(0\). But \(231\) is not divisible by \(10\) because it ends in \(1\).

Meaning: If a number ends in \(0\), it is divisible by \(10\). No other ending works.

Divisibility Rule for 3 and 9

A number is divisible by \(3\) if the sum of its digits is divisible by \(3\). A number is divisible by \(9\) if the sum of its digits is divisible by \(9\). This means you add the digits first, then check the result.

These rules are helpful because they work even for large numbers. They are often seen in school lessons and exam reviews because they test number sense, not just memorization.

Shortcut: Add all digits quickly. If the total is a multiple of \(3\), the number is divisible by \(3\). If the total is a multiple of \(9\), the number is divisible by \(9\).

Example: \(372\) is divisible by \(3\) because \(3 + 7 + 2 = 12\), and \(12\) is divisible by \(3\). It is not divisible by \(9\) because \(12\) is not divisible by \(9\).

Meaning: Add the digits. If the sum divides by \(3\) or \(9\), then the original number also divides by \(3\) or \(9\).

Divisibility Rule for 4 and 8

A number is divisible by \(4\) if its last two digits are divisible by \(4\). A number is divisible by \(8\) if its last three digits are divisible by \(8\). These rules help you check larger numbers more easily.

This is helpful when the number is too big for quick mental division. Civil Service Exam practice items often use these rules because they are fast ways to find exact divisibility.

Tip: For \(4\), check only the last two digits. For \(8\), check only the last three digits. You do not need the whole number.

Example: \(1,236\) is divisible by \(4\) because the last two digits are \(36\), and \(36\) is divisible by \(4\). The number \(9,208\) is divisible by \(8\) because the last three digits are \(208\), and \(208\) is divisible by \(8\).

Meaning: For \(4\), look at the last two digits, and for \(8\), look at the last three digits. If those parts divide evenly, the whole number does too.

Divisibility Rule for 6

A number is divisible by \(6\) if it is divisible by both \(2\) and \(3\). This means the number must be even, and the sum of its digits must also be divisible by \(3\). Both conditions must be true.

This rule combines two earlier rules, so it is a good test of careful thinking. It is often used in school and exam questions because it checks whether you can apply more than one rule at the same time.

Tip: First check if the number is even. Then add the digits. If both pass, the number is divisible by \(6\).

Example: \(84\) is divisible by \(6\) because it is even and \(8 + 4 = 12\), which is divisible by \(3\). But \(82\) is not divisible by \(6\) because \(8 + 2 = 10\), and \(10\) is not divisible by \(3\).

Meaning: A number must pass the rules for \(2\) and \(3\) before it can pass the rule for \(6\).

Divisibility Rule for 11

A number is divisible by \(11\) if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is \(0\) or a multiple of \(11\). This may sound hard at first, but it becomes easier with practice.

This rule is sometimes used in higher-level number exercises and exam review materials. It is helpful when you want a quick check on longer numbers without full division.

Shortcut: For a quick check, add alternating digits and subtract the two totals. If the result is \(0\) or a multiple of \(11\), the number is divisible by \(11\).

Example: In \(121\), the first and third digits sum to \(2\), and the middle digit is \(2\), so the difference is \(0\). That means \(121\) is divisible by \(11\).

Meaning: Split the digits into two groups, compare their sums, and see if the difference is \(0\) or \(11\).

Remainders

A remainder is what is left after division when the number does not divide evenly. If a number is not divisible by another, there will be a remainder. For example, \(17 \div 5 = 3\) remainder \(2\).

Remainders are important because they tell you how much is left after sharing or grouping. They are also common in test questions that ask what remains after division.

Tip: The remainder is always smaller than the divisor. If you divide by \(6\), the remainder can only be \(0\), \(1\), \(2\), \(3\), \(4\), or \(5\).

Example: \(19 \div 4 = 4\) remainder \(3\). This means \(4\) groups of \(4\) can be made, and \(3\) is left over.

Meaning: The remainder is the leftover part after dividing. If there is no leftover, the remainder is \(0\).

Divisibility and Remainders Together

Divisibility and remainders are connected. If a number is divisible by another number, the remainder is \(0\). If it is not divisible, then the remainder is any number smaller than the divisor.

This connection is useful in solving word problems and exam items. It helps you decide quickly whether to use exact division or to look for what is left.

Shortcut: If you want the remainder, divide only until you find the nearest smaller multiple. Then subtract. For example, \(25 \div 6\): the nearest smaller multiple of \(6\) is \(24\), so the remainder is \(1\).

Example: \(24 \div 6 = 4\), so \(24\) is divisible by \(6\) and the remainder is \(0\). But \(25 \div 6 = 4\) remainder \(1\), so \(25\) is not divisible by \(6\).

Meaning: Divisible means no remainder. Not divisible means there is leftover.

Tips and Shortcuts

When you work with big numbers, do not solve everything by long division first. Use divisibility rules to save time. This is the fastest way to answer many test items because you only check a small part of the number.

The best habit is to start with the easiest rules first. Check the last digit, then the last two digits, then the sum of digits. Many big-number questions become simple when you break them into smaller parts.

Tips:

• Check \(2\), \(5\), and \(10\) first because they are the easiest.
• For \(3\) and \(9\), add the digits quickly.
• For \(4\) and \(8\), only check the last \(2\) or \(3\) digits.
• For \(6\), make sure the number passes both \(2\) and \(3\).
• For \(11\), use alternating digit sums.

Big Number Trick: If a number is very long, divide it into small parts in your mind. Look at the ending digits first. Most divisibility questions can be answered without writing the whole solution.

Example: To check \(4,782\) for divisibility by \(4\), look at \(82\). Since \(82\) is not divisible by \(4\), the whole number is not divisible by \(4\).

Example: To check \(7,236\) for divisibility by \(3\), add \(7 + 2 + 3 + 6 = 18\). Since \(18\) is divisible by \(3\), the whole number is divisible by \(3\).

Example: To check \(52,184\) for divisibility by \(8\), look at \(184\). Since \(184\) is divisible by \(8\), the whole number is divisible by \(8\).

More Practice Examples

Here are more examples to help you move faster. The more you practice, the quicker your brain will recognize the rule. That is why short checks are very useful before doing full division.

Example 1: \(450\) is divisible by \(2\), \(5\), and \(10\) because it ends in \(0\).

Example 2: \(729\) is divisible by \(9\) because \(7 + 2 + 9 = 18\), and \(18\) is divisible by \(9\).

Example 3: \(1,016\) is divisible by \(4\) because the last two digits are \(16\), and \(16\) is divisible by \(4\).

Example 4: \(3,456\) is divisible by \(8\) because the last three digits are \(456\), and \(456\) is divisible by \(8\).

Example 5: \(1,243\) is not divisible by \(3\) because \(1 + 2 + 4 + 3 = 10\), and \(10\) is not divisible by \(3\).

What To Remember

• A number is divisible when it has no remainder after division.
• For \(2\), check the last digit if it is even.
• For \(5\), the last digit must be \(0\) or \(5\).
• For \(10\), the last digit must be \(0\).
• For \(3\) and \(9\), add the digits first.
• For \(4\), check the last two digits.
• For \(8\), check the last three digits.
• For \(6\), the number must be divisible by both \(2\) and \(3\).
• For \(11\), compare digit sums in alternating positions.
• A remainder is the leftover after division.

Multiple Choice Questions

Answer these in the comment section. Choose the best answer for each item.

1. Which number is divisible by \(5\)?

A. \(124\)
B. \(135\)
C. \(248\)
D. \(317\)

2. Which number is divisible by \(3\)?

A. \(214\)
B. \(342\)
C. \(517\)
D. \(901\)

3. Which number is divisible by \(4\)?

A. \(1,234\)
B. \(1,236\)
C. \(1,238\)
D. \(1,242\)

4. Which number is divisible by \(6\)?

A. \(72\)
B. \(73\)
C. \(75\)
D. \(77\)

5. What is the remainder when \(29\) is divided by \(4\)?

A. \(1\)
B. \(2\)
C. \(3\)
D. \(4\)

Write this on your notes.

• Divisibility - A number can be divided evenly with no remainder.
• Remainder - The leftover after division.
• Divisible by 2 - The last digit is 0, 2, 4, 6, or 8.
• Divisible by 5 - The last digit is 0 or 5.
• Divisible by 10 - The last digit is 0.
• Divisible by 3 - The sum of the digits is divisible by 3.
• Divisible by 9 - The sum of the digits is divisible by 9.
• Divisible by 4 - The last two digits are divisible by 4.
• Divisible by 8 - The last three digits are divisible by 8.
• Divisible by 6 - The number is divisible by both 2 and 3.
• Divisible by 11 - Compare digit sums in alternating positions.
• Exact division - Division with no remainder.
• Leftover - The part that remains after dividing.
• Even number - A number that ends in 0, 2, 4, 6, or 8.
• Odd number - A number that ends in 1, 3, 5, 7, or 9.