Divisibility Rules

To solve this problem, we need to determine the relationship between and when we know that divides .

1. The Condition Given:
The problem states that divides , i.e., . This means that is divisible by . According to number theory, if a prime divides the square of a number, then must divide the number itself.

2. Applying the Rule:
Since , we can conclude that must also divide (i.e., ). This is because if a prime divides the square of a number, it divides the number itself.

3. Evaluating the Options:

• (A) divides – Correct ✔; as explained, if , then .
• (B) divides – Incorrect; we cannot conclude that divides from the given condition.
• (C) divides – Incorrect; the condition does not imply this.
• (D) None – Incorrect; the correct answer is option (A).

Final Answer:
The correct answer is (A) 7 divides .

To solve the problem, we need to count how many two-digit numbers are divisible by 3.

1. Identifying the Range:
Two-digit numbers range from 10 to 99.

2. Finding the First and Last Two-Digit Numbers Divisible by 3:
First two-digit number divisible by 3 = 12
Last two-digit number divisible by 3 = 99

3. Applying the Arithmetic Progression (AP) Formula:
This forms an AP where:
First term,
Last term,
Common difference,

4. Using the AP formula:
The number of terms in an AP is given by:



Substitute values:

Final Answer:
The number of two-digit numbers divisible by 3 is 30.