Answer: Option B. -> 9.6 m

Answer: Option C. -> 21

Answer: Option D. -> 3640 second

Answer: Option B. -> 4

Answer: Option D. -> 24
To solve this problem, we need to find the greatest common divisor (GCD) of the three given numbers. The GCD is the largest number that divides all the given numbers without leaving any remainder. However, in this case, we are also given that the number should leave the same remainder when it divides each number.
Here are the steps to solve the problem:
Step 1: Find the difference between any two of the given numbers.Difference between 30 and 78 = 78 - 30 = 48Difference between 78 and 102 = 102 - 78 = 24Difference between 102 and 30 = 102 - 30 = 72
Step 2: Find the GCD of the differences.GCD(48, 24, 72) = 24
Step 3: The GCD of the differences is the required answer.Therefore, the largest number which divides 30, 78 and 102 to leave the same remainder in each case is 24.
Explanation:

• GCD: The greatest common divisor (GCD) of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder.
• Remainder: The remainder is the amount left over after division. For example, when we divide 7 by 3, the quotient is 2 with a remainder of 1.
• Difference: The difference between two numbers is the result of subtracting one number from the other.
• Euclidean Algorithm: The Euclidean algorithm is a method for finding the GCD of two numbers. The algorithm states that the GCD of two numbers is equal to the GCD of the smaller number and the remainder of the larger number divided by the smaller number.

In this problem, we used the fact that if a number leaves the same remainder when it divides three given numbers, then it must also leave the same remainder when it divides the differences between those numbers. By finding the GCD of the differences, we were able to find the GCD of the original numbers. The final answer is 24, which is the largest number that satisfies the given conditions.

Answer: Option A. -> 10080

Answer: Option C. -> 1394

Answer: Option C. -> 63 min.
Let x be the greatest unit of time that can divide both 5 hours 15 minutes and 8 hours 24 minutes into integers.
We can write 5 hours 15 minutes as 5x + 15y, where x is the unit of time and y is the fraction of x. Similarly, 8 hours 24 minutes can be written as 8x + 24y.
We want both expressions to be integers, which means y must also be an integer.
To find the greatest unit of time that can divide both expressions into integers, we need to find the greatest common factor (GCF) of the two expressions.
Let's first convert the minutes to hours:5 hours 15 minutes = 5.25 hours8 hours 24 minutes = 8.4 hours
Now, we can write:5x + 15y = 5.25k ...(1)8x + 24y = 8.4k ...(2)
where k is a positive integer (since both expressions must be integers).
Let's simplify these expressions by multiplying them by 100 to get rid of the decimals:
500x + 1500y = 525k ...(3)800x + 2400y = 840k ...(4)
To find the GCF of these two expressions, we can subtract (3) from (4):
300x + 900y = 315k
We can see that 300 and 900 have a common factor of 300, so we can simplify the expression further:
x + 3y = 3.15k
Since x and y are integers, 3y must be a multiple of 3.
We can also see that x and 3y have a common factor of 3, so let's write:
x = 3m3y = 3n
where m and n are integers.
Substituting these values in the equation above, we get:
3m + 3n = 3.15km + n = 1.05k
We want to find the greatest unit of time, which means we want to find the greatest value of k for which m and n are integers.
The greatest common factor of 105 and 100 is 5.
So, let's try k = 5:
m + n = 5.25
The only integer values of m and n that satisfy this equation are:
m = 2n = 3
Substituting these values in x and y, we get:
x = 6y = 9
So, the greatest unit of time with which both 5 hours 15 minutes and 8 hours 24 minutes can be represented as integers is 63 minutes (Option C).If you think the solution is wrong then please provide your own solution below in the comments section .

Answer: Option C. -> 15

Answer: Option B. -> 80