Understanding Divisibility Rules

Divisibility rules are handy shortcuts that help us determine if a number can be evenly divided by another number without performing long division. Understanding these rules is fundamental in number theory and makes calculations easier. Let's examine the given statements about divisibility rules to see which ones are correct.

Analyzing Each Divisibility Statement

Statement (A): Divisibility by 6

The statement says: "A number is divisible by 6 if it is divisible by 3 or 2."

The correct rule for divisibility by 6 is that a number must be divisible by both 2 and 3. Divisibility by 2 means the number is even (ends in 0, 2, 4, 6, or 8). Divisibility by 3 means the sum of its digits is divisible by 3.

Let's consider some examples for the statement:

• Consider the number 9. It is divisible by 3 (\(9 \div 3 = 3\)). It is not divisible by 2 (it's odd). According to statement (A), since it's divisible by 3 (the "or" condition), it should be divisible by 6. However, 9 is not divisible by 6 (\(9 \div 6 \approx 1.5\)).
• Consider the number 4. It is divisible by 2 (\(4 \div 2 = 2\)). It is not divisible by 3 (\(4 \div 3 \approx 1.33\)). According to statement (A), since it's divisible by 2 (the "or" condition), it should be divisible by 6. However, 4 is not divisible by 6 (\(4 \div 6 \approx 0.67\)).

Since numbers divisible by 3 OR 2 are not necessarily divisible by 6, statement (A) is incorrect. It should be "if and only if it is divisible by 3 and 2".

Statement (B): Divisibility by 5

The statement says: "A number is divisible by 5 if its unit digit is 0 or 5."

The correct rule for divisibility by 5 states that a number is divisible by 5 if its last digit (the unit digit) is either 0 or 5.

Let's check with examples:

• The number 25 ends with 5. \(25 \div 5 = 5\). It is divisible by 5.
• The number 120 ends with 0. \(120 \div 5 = 24\). It is divisible by 5.
• The number 32 ends with 2. It does not end with 0 or 5. 32 is not divisible by 5 (\(32 \div 5 = 6.4\)).

This rule is accurate and widely used. Statement (B) is correct.

Statement (C): Divisibility by 3

The statement says: "A number is divisible by 3 if its unit digit is divisible by 3."

The correct rule for divisibility by 3 states that a number is divisible by 3 if the sum of its digits is divisible by 3.

Let's check the statement with examples:

• Consider the number 13. Its unit digit is 3. The unit digit 3 is divisible by 3 (\(3 \div 3 = 1\)). According to statement (C), 13 should be divisible by 3. However, 13 is not divisible by 3 (\(13 \div 3 \approx 4.33\)).
• Consider the number 12. Its unit digit is 2. The unit digit 2 is not divisible by 3. According to statement (C), 12 should not be divisible by 3. However, 12 is divisible by 3 (\(12 \div 3 = 4\)).

Statement (C) describes an incorrect rule for divisibility by 3. The unit digit alone is not sufficient; the sum of all digits is required.

Statement (D): Divisibility by 4

The statement says: "A number is divisible by 4 if the number formed by its last two digits is divisible by 4."

The correct rule for divisibility by 4 states that a number is divisible by 4 if the number formed by its last two digits is divisible by 4. This rule applies regardless of how many digits the number has.

Let's check with examples:

• Consider the number 116. The last two digits form the number 16. 16 is divisible by 4 (\(16 \div 4 = 4\)). According to statement (D), 116 should be divisible by 4. \(116 \div 4 = 29\). It is divisible by 4.
• Consider the number 532. The last two digits form the number 32. 32 is divisible by 4 (\(32 \div 4 = 8\)). According to statement (D), 532 should be divisible by 4. \(532 \div 4 = 133\). It is divisible by 4.
• Consider the number 718. The last two digits form the number 18. 18 is not divisible by 4 (\(18 \div 4 = 4.5\)). According to statement (D), 718 should not be divisible by 4. \(718 \div 4 = 179.5\). It is not divisible by 4.

Statement (D) describes the correct rule for divisibility by 4. Statement (D) is correct.

Summary of Correct Divisibility Statements

Based on our analysis of each statement:

• Statement (A) is incorrect.
• Statement (B) is correct.
• Statement (C) is incorrect.
• Statement (D) is correct.

The correct statements are (B) and (D) only.

Selecting the Correct Option

We need to find the option that lists only statements (B) and (D) as correct.

• Option (a) says (A), (B), and (D) only. This includes the incorrect statement (A).
• Option (b) says (B) and (D) only. This includes only the correct statements (B) and (D).
• Option (c) says (A), (C), and (D) only. This includes the incorrect statements (A) and (C).
• Option (d) says (B) and (C) only. This includes the incorrect statement (C).

The option that correctly identifies the true statements about divisibility is (b), which lists (B) and (D) only.

Revision Table: Divisibility Rules Summary

Here is a summary of some common divisibility rules:

Additional Information: Number Properties

Understanding divisibility rules connects to other important number properties:

• Factors: If a number \(a\) is divisible by a number \(b\), then \(b\) is a factor of \(a\). For example, since 12 is divisible by 3, 3 is a factor of 12.
• Multiples: If a number \(a\) is divisible by a number \(b\), then \(a\) is a multiple of \(b\). For example, since 12 is divisible by 3, 12 is a multiple of 3.
• Prime Numbers: A prime number is a whole number greater than 1 that has only two factors: 1 and itself. Divisibility rules help determine if a number has factors other than 1 and itself.
• Composite Numbers: A composite number is a whole number greater than 1 that has more than two factors.

These concepts are all interlinked and form the basis of number theory.