The problem asks for the greatest number that divides 104 and 133, leaving specific remainders. This means that if we subtract the remainders from the original numbers, the resulting numbers must be perfectly divisible by the greatest number we are looking for. Therefore, we need to find the HCF (Highest Common Factor) of these new numbers.
When 104 is divided by the required number, the remainder is 8. This means that \((104 - 8)\) must be perfectly divisible by the number.
\[ 104 - 8 = 96 \]When 133 is divided by the required number, the remainder is 5. This means that \((133 - 5)\) must be perfectly divisible by the number.
\[ 133 - 5 = 128 \]Now we need to find the HCF of 96 and 128. We can use the prime factorization method or the Euclidean algorithm.
\[ 96 = 2 \times 48 = 2 \times 2 \times 24 = 2 \times 2 \times 2 \times 12 = 2 \times 2 \times 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 = 2^5 \times 3^1 \]
\[ 128 = 2 \times 64 = 2 \times 2 \times 32 = 2 \times 2 \times 2 \times 16 = 2 \times 2 \times 2 \times 2 \times 8 = 2 \times 2 \times 2 \times 2 \times 2 \times 4 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^7 \]
The HCF is the product of the common prime factors raised to the lowest power they appear in either factorization.
Common prime factor is 2. The lowest power of 2 is \(2^5\).
\[ \text{HCF}(96, 128) = 2^5 = 32 \]The number we found, 32, must be greater than the remainders (8 and 5). Since \(32 > 8\) and \(32 > 5\), this is a valid solution.
B) 32 is the greatest number that divides 104 and 133, leaving remainders 8 and 5 respectively. When 104 is divided by 32, \(104 = 3 \times 32 + 8\). When 133 is divided by 32, \(133 = 4 \times 32 + 5\).
The question asks to identify the city that hosted a specific international conference in November 2024. This requires knowledge of recent global events and their locations, particularly those involving international organizations like the International Cooperative Alliance (ICA).
C) New Delhi — New Delhi is indeed the host city for the International Cooperative Alliance (ICA) Global Cooperative Conference and General Assembly scheduled for November 2024. This event is significant for the cooperative movement globally, and India's capital was chosen as the venue.
The mode of a set of observations is the value that appears most frequently in the data set. To find the mode, we need to count the occurrences of each distinct observation and identify the one with the highest frequency.
3, 6, 5, 7, 8, 4, 3, 3, 2, 5, 3, 6, 4, 8, 5
Comparing the frequencies:
The highest frequency is 4, which corresponds to the observation 3.
Since 3 appears most frequently (4 times), the mode of the given observations is 3.
Correct Option: D) 3
We need to find the length of the longer side of a rectangle given two conditions: the difference between its sides and a relationship between its area and perimeter. We will set up algebraic equations based on these conditions and solve for the dimensions.
Correct Option: D) 36
To find the effective price percentage after consecutive discounts, we can calculate the remaining percentage of the marked price after each discount. Alternatively, we can use a formula for successive discounts or calculate the final price by assuming an initial marked price of 100.
B) 74.96% is the correct effective price percentage after applying all three consecutive discounts, rounded to two decimal places.
The problem asks us to find the next number in the given series: 5, 8, 17, 44, 125, ?. We need to identify the pattern or rule governing the sequence.
D) 368 is the correct answer. The pattern involves adding successive powers of 3 (or multiples of 3) to the previous term: \(5 + 3^1 = 8\), \(8 + 3^2 = 17\), \(17 + 3^3 = 44\), \(44 + 3^4 = 125\). Therefore, the next term is \(125 + 3^5 = 125 + 243 = 368\).
An electric pump can fill a tank in 6 hours. Due to a leakage in the tank, it takes 7 \( \frac{1}{5} \) hours to fill the tank. How much time will this leak take to empty the full tank if water does not get in or out of the tank through any other point during this period?
This problem involves concepts of work and time, specifically dealing with rates of filling and emptying a tank. We can determine the individual rates of the pump and the leak, and then use these rates to find the time taken by the leak alone to empty the tank.
Calculate the rate of the pump: The pump fills the tank in 6 hours. So, in 1 hour, the pump fills \(\frac{1}{6}\) of the tank.
\[ \text{Pump's rate} = \frac{1}{6} \text{ tank/hour} \]Calculate the combined rate of the pump and the leak: Due to the leak, the tank is filled in \(7 \frac{1}{5}\) hours. Convert this mixed fraction to an improper fraction:
\[ 7 \frac{1}{5} = \frac{(7 \times 5) + 1}{5} = \frac{35 + 1}{5} = \frac{36}{5} \text{ hours} \]So, in 1 hour, the pump (with the leak) fills \[\frac{1}{\frac{36}{5}}\] of the tank, which is \(\frac{5}{36}\) of the tank.
\[ \text{Combined rate (Pump - Leak)} = \frac{5}{36} \text{ tank/hour} \]Calculate the rate of the leak: Let the rate of the leak be \(x\) tank/hour. The combined rate is the pump's rate minus the leak's rate (since the leak empties water).
\[ \text{Pump's rate} - \text{Leak's rate} = \text{Combined rate} \] \[ \frac{1}{6} - x = \frac{5}{36} \]Now, solve for \(x\):
\[ x = \frac{1}{6} - \frac{5}{36} \]To subtract these fractions, find a common denominator, which is 36:
\[ x = \frac{6}{36} - \frac{5}{36} \] \[ x = \frac{1}{36} \text{ tank/hour} \]This means the leak empties \(\frac{1}{36}\) of the tank in 1 hour.
Calculate the time taken by the leak to empty the full tank: If the leak empties \(\frac{1}{36}\) of the tank in 1 hour, then to empty the full tank (which is 1 unit), it will take the reciprocal of this rate.
\[ \text{Time taken by leak} = \frac{1}{\text{Leak's rate}} = \frac{1}{\frac{1}{36}} = 36 \text{ hours} \]Correct Option: C) 36 hours
The problem presents a relationship between letter-clusters. We need to identify the pattern in the given pair (TNJ : %) and apply the same pattern to find the missing cluster (# : WQM).
Let's assume the first pair is ZTP : WQM. We need to find the pattern from ZTP to WQM.
The pattern is a subtraction of 3 from the alphabetical position of each letter.
Now, let's apply this pattern to TNJ to find %.
So, TNJ becomes QKG. This matches the given % = QKG in Option A.
Since both parts of Option A follow the same pattern, this is likely the correct answer.
A) # = ZTP, % = QKG
The pattern observed is that each letter in the first cluster is replaced by a letter whose alphabetical position is 3 less than the original letter's position.
Both pairs follow the same -3 pattern.
The question asks for the correct method to create a new folder on the Desktop in Windows 11. We need to identify the standard graphical user interface (GUI) action or keyboard shortcut that performs this specific task.
Correct Option: D) Right-click → Select New → Folder
This sequence of actions is the standard and most direct way to create a new folder on the Desktop in Windows 11.