Divisibility And Remainder

Let the two consecutive even integers be and . Then, their squares will be and . According to the question: Divide the whole equation by 2: Solve the quadratic: So, or Only the positive even integer is valid:

Thus, the numbers are 12 and 14

We are tasked with finding the set difference , where:

Step 1: Recall the definition of set difference.

The set difference is the set of elements that belong to but do not belong to . Mathematically:

Step 2: Identify the elements in that are not in .

The elements of are , and the elements of are . The elements and are common to both sets, so they are excluded from .

The remaining elements in are .

Step 3: Write the result.

Final Answer: The set difference is , which corresponds to option .

Step 1: Recall the relationship between the characteristic of a logarithm and the number of digits.

The characteristic of the logarithm of a number gives information about the order of magnitude of the number. Specifically:

• If the characteristic of is , then the number lies in the range:
• A number in this range has digits. This is because the smallest number with digits is , and the largest number with digits is .

Step 2: Match the relationship to the options.

If the characteristic of is , the number of digits in is .

Final Answer: The number of digits in the number is , which corresponds to option .

Step 1: Analyze each statement.

(1) Logarithm of 1 to any non-zero base is 0.

The logarithmic identity states that for any base and :

This is because . Hence, this statement is true.

(2) Logarithm of any non-zero number to the same base is 1.

The logarithmic identity states that for any base and :

This is because . Hence, this statement is true.

(3) Logarithms of a number with different bases have different values.

The value of a logarithm depends on both the number and the base. For example:

Thus, the logarithm of the same number can indeed have different values depending on the base. This statement is true.

(4) All of the above.

Since all three statements (1), (2), and (3) are true, this option is true.

Final Answer: The correct option is , which corresponds to option .

Step 1: Express the problem mathematically.

Let the number be . Then:

Step 2: Rewrite the congruences.

We can rewrite the congruences as:

Step 3: Find the least common multiple (LCM) of 28 and 32.

The LCM of 28 and 32 is:

Step 4: Solve for the smallest .

The general solution is:

where satisfies both congruences. Testing values, we find:

Final Answer: The smallest number is , which corresponds to option .

Given that and , where and are prime numbers, we need to find the HCF of and . The prime factorization of is: The prime factorization of is: To find the HCF, we take the lowest powers of the common prime factors. The common prime factors are and . For , the lowest power is , and for , the lowest power is . Thus, the HCF of and is:

The correct option is (B):

We are given the equation Using the property of logarithms and similarly, , we can rewrite the given equation as: Simplifying: Using the change of base formula , we get: Thus, the equation becomes: which simplifies to: Therefore,

So. the correct option is (A): a+b=1

According to the fundamental theorem of arithmetic, if a prime divides , it must divide . This is because if a prime divides the square of a number, it divides the number itself. Therefore, the correct answer is that divides .

The correct option is (C): p divides a

All prime numbers greater than 3 can be expressed in one of two forms:

or
(where is a positive integer)

Case 1:

When divided by 6, the remainder is 1.

Case 2:

When divided by 6, the remainder is again 1.

Verification with Examples

leaves remainder 1

leaves remainder 1

leaves remainder 1

Conclusion

For any prime number , .
The correct answer is (1) 1.