All questions of Playing with Numbers for Class 6 Exam

A prime number must have exactly two factors (1 and itself). Therefore, the number 2 is the lowest prime number as its only common divisors are 1 and 2. The number 1 only has 1 common divisor so it cannot be a prime number.

Divisibility Rules
- Divisibility rule of 2: If a number is even, it is divisible by 2.
- Divisibility rule of 3: If the sum of the digits of a number is divisible by 3, then the number is divisible by 3.
- Divisibility rule of 6: If a number is divisible by both 2 and 3, then it is divisible by 6.

Explanation
- If a number is divisible by 2, it means it is an even number.
- If the sum of the digits of a number is divisible by 3, it means the number is divisible by 3.
- If a number is divisible by both 2 and 3, it means the number is even and the sum of its digits is also divisible by 3.
- Therefore, the number is also divisible by 6.

Conclusion
- Hence, if a number is divisible by 2 and 3 both, then it is divisible by 6.
- Therefore, the correct answer is option 'B'.

B

To determine the numbers by which a given number will be divisible, we need to consider the factors of that number. In this case, we are given that the number is divisible by 12.

Factors of 12:
The factors of 12 are the numbers that divide 12 without leaving a remainder. These factors are 1, 2, 3, 4, 6, and 12.

Divisibility by 3:
To determine if a number is divisible by 3, we need to check if the sum of its digits is divisible by 3. In the case of 12, the sum of its digits (1 + 2) is 3, which is divisible by 3. Therefore, 12 is divisible by 3.

Divisibility by 4:
To determine if a number is divisible by 4, we need to check if the last two digits form a number that is divisible by 4. In the case of 12, the last two digits are 12 itself, which is divisible by 4. Therefore, 12 is divisible by 4.

So, the numbers by which 12 will be divisible are 1, 2, 3, 4, 6, and 12.

Out of the given options, the correct answer is option 'D' (3 and 4) because both 3 and 4 are factors of 12.

The correct answer is option 'C': multiple.

Explanation:
To understand why the correct answer is 'multiple', let's first define what a multiple is. A multiple of a number is the result of multiplying that number by an integer (whole number). For example, the multiples of 3 are 3, 6, 9, 12, 15, and so on.

Now, let's consider the given statement: "Every ______ of a number is greater than or equal to that number." This statement implies that whatever fills the blank, it should be something that, when multiplied by a number, the result is greater than or equal to the original number.

Let's consider the options provided:
a) Factor: A factor of a number is a whole number that divides the given number without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. However, if we multiply a number by one of its factors, the result may or may not be greater than the original number. For instance, if we multiply 12 by 2, we get 24, which is greater than 12, but if we multiply 12 by 3, we get 36, which is greater than 12. Therefore, a factor is not always greater than or equal to the original number.

b) Number: This option is incorrect because it is too broad. The statement is already referring to a number, so it doesn't provide any additional information.

c) Multiple: A multiple of a number, as mentioned earlier, is obtained by multiplying the number by an integer. Since we are multiplying the number, the resulting multiple will always be greater than or equal to the original number. For example, if we multiply 5 by 2, we get 10, which is greater than 5. Similarly, if we multiply 5 by 1, we get 5, which is equal to the original number. Therefore, every multiple of a number is greater than or equal to that number.

d) None of these: This option is incorrect because the correct answer is 'C' - multiple.

In conclusion, the correct answer is 'C' - multiple because every multiple of a number is greater than or equal to that number.

If a number is divisible by 10, then its one's digit can only be 0. This is because any number divisible by 10 must end in 0.

HCF of two consecutive numbers is always 1 . explanation : Take two consecutive numbers as 9 and 10. between these two numbers there is no common factor other than 1. Take two consecutive numbers as 17 and 18.

We know 3 x 3 = 9. So The exact divisor of number 9.

Solution:

To understand the solution, we need to know the divisibility rules of 6.

Divisibility Rule of 6: A number is divisible by 6 if it is divisible by both 2 and 3.

Now, let us consider each option one by one.

a) 3 and 4: If a number is divisible by 3 and 4, it doesn't mean that it is divisible by 6. For example, the number 12 is divisible by 3 and 4, but it is also divisible by 6. However, the number 8 is divisible by 4 but not by 6. Hence, option a) is incorrect.

b) 2 and 5: If a number is divisible by 2 and 5, it doesn't mean that it is divisible by 6. For example, the number 10 is divisible by 2 and 5, but it is not divisible by 6. Hence, option b) is incorrect.

c) 2 and 3: According to the divisibility rule of 6, if a number is divisible by 2 and 3, it is also divisible by 6. Hence, option c) is correct.

d) None of these: Since option c) is correct, option d) is incorrect.

Therefore, the correct option is c) 2 and 3.

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