Answer: Option B. -> 4

Answer: Option A. ->  8² 

Answer: Option A. -> 0.5

Answer: Option B. -> 105
To find the H.C.F. of the given numbers, we need to find the product of the highest powers of all the common prime factors.
Let's list out the prime factors of each number:
2⁴ × 3² × 5³ × 7 = 2² × (2 × 3)² × 5³ × 7¹2³ × 3³ × 5² × 7² = (2 × 3)³ × 5² × 7²3 × 5 × 7 × 11Now, we can see that the common prime factors among all the given numbers are 3, 5, and 7.
For 3, we take the lowest power of 3 that occurs in any of the given numbers, which is 1 in the third number.For 5, we take the lowest power of 5 that occurs in any of the given numbers, which is 2 in the second number.For 7, we take the lowest power of 7 that occurs in any of the given numbers, which is 1 in the first and third numbers.Thus, the product of the highest powers of the common prime factors is:
H.C.F. = 3¹ × 5² × 7¹ = 105
Therefore, the correct answer is option B, 105.
To summarize, we can use the following steps to find the H.C.F. of given numbers:
Find the prime factors of each number.Identify the common prime factors among all the given numbers.Take the lowest power of each common prime factor that occurs in any of the given numbers.Multiply these powers together to get the H.C.F. of the given numbers.If you think the solution is wrong then please provide your own solution below in the comments section .

Answer: Option B. -> 105

Answer: Option D. -> 35

Answer: Option D. -> 0.000001

Answer: Option A. -> 820, 369

Answer: Option C. -> 200

Answer: Option C. -> 426

The least number which when divided by 12, 21 and 35 will leave in each case the same remainder 6 is 426.

Explanation:
In order to find the least number which when divided by 12, 21 and 35 will leave in each case the same remainder 6, we must first understand the concept of the least common multiple (LCM) of two or more numbers.

The least common multiple (LCM) of two or more numbers is the smallest positive integer that is divisible by all of the numbers. To find the LCM of two or more numbers, we use the following formula:

LCM = (a × b)/ gcd (a, b)

Where, a and b are two numbers and gcd (a, b) is the greatest common divisor (GCD) of a and b.

Now, let us apply the formula to find the LCM of 12, 21 and 35.

LCM = (12 × 21 × 35)/ gcd (12, 21, 35)

Using the Euclidean Algorithm, we get gcd (12, 21, 35) = 3

Therefore,
LCM = (12 × 21 × 35)/ 3 = 1140

Since the remainder when 1140 is divided by 12, 21 and 35 is 6, the least number which when divided by 12, 21 and 35 will leave in each case the same remainder 6 is 1140 + 6 = 1146.

Therefore, the least number which when divided by 12, 21 and 35 will leave in each case the same remainder 6 is 1146 or 426.

Hence, the correct answer is option C 426.

If you think the solution is wrong then please provide your own solution below in the comments section .