To find the smallest six-digit number completely divisible by 4, 8, 12, and 16, we first need to find the Least Common Multiple (LCM) of these numbers. Then, we'll find the smallest six-digit number and determine the smallest multiple of the LCM that is greater than or equal to this six-digit number.
Find the LCM of 4, 8, 12, and 16:
First, write down the prime factorization of each number:
The LCM is found by taking the highest power of all prime factors present:
\[ \text{LCM}(4, 8, 12, 16) = 2^4 \times 3^1 = 16 \times 3 = 48 \]Identify the smallest six-digit number:
The smallest six-digit number is \(100000\).
Divide the smallest six-digit number by the LCM:
Divide \(100000\) by \(48\):
\[ 100000 \div 48 \]Performing the division:
\[ 100000 = 48 \times 2083 + 16 \]Here, the quotient is \(2083\) and the remainder is \(16\).
Find the smallest six-digit multiple of the LCM:
To find the smallest six-digit number completely divisible by 48, we need to add the difference between the LCM and the remainder to the smallest six-digit number. If the remainder is 0, the number itself is divisible. Since the remainder is 16, we need to add \((48 - 16)\) to \(100000\).
\[ \text{Required number} = 100000 + (48 - 16) \] \[ \text{Required number} = 100000 + 32 \] \[ \text{Required number} = 100032 \]Alternatively, the next multiple of 48 after \(48 \times 2083\) is \(48 \times (2083 + 1)\):
\[ 48 \times 2084 = 100032 \]A) 100032 is the smallest six-digit number completely divisible by 4, 8, 12, and 16.
To find the new ratio of male to female members, we first need to determine the initial number of male and female members based on the given ratio. Then, we calculate the increased number of males and females by applying the respective percentage increases. Finally, we form the ratio of these new numbers and simplify it.
Let the initial number of male members be \(4x\) and female members be \(5x\), based on the given ratio of 4:5.
The number of males is increased by 80%.
\[ \text{Increase in males} = 80\% \text{ of } 4x = \frac{80}{100} \times 4x = \frac{4}{5} \times 4x = \frac{16x}{5} \]New number of males = Initial males + Increase in males
\[ \text{New males} = 4x + \frac{16x}{5} = \frac{20x + 16x}{5} = \frac{36x}{5} \]The number of females is increased by 60%.
\[ \text{Increase in females} = 60\% \text{ of } 5x = \frac{60}{100} \times 5x = \frac{3}{5} \times 5x = 3x \]New number of females = Initial females + Increase in females
\[ \text{New females} = 5x + 3x = 8x \]The new ratio of male members to female members is:
\[ \frac{\text{New males}}{\text{New females}} = \frac{\frac{36x}{5}}{8x} = \frac{36x}{5 \times 8x} = \frac{36}{40} \]Divide both numerator and denominator by their greatest common divisor, which is 4.
\[ \frac{36 \div 4}{40 \div 4} = \frac{9}{10} \]The new ratio of male members to female members is 9:10.
B) 9:10 is the correct ratio after the specified increases in male and female members.
The question asks to identify the scientist who proposed the three laws of motion. This is a direct knowledge-based question related to classical mechanics.
A) Newton — Sir Isaac Newton published his three laws of motion in his work "Philosophiæ Naturalis Principia Mathematica" (Mathematical Principles of Natural Philosophy) in 1687. These laws are: the law of inertia, the law of acceleration (\(F=ma\)), and the law of action and reaction.
The question asks to identify the cyclone that affected Odisha and West Bengal in May 2020. This requires knowledge of major cyclonic events in India, specifically their names and the year/month of their occurrence.
Super Cyclonic Storm Amphan was a powerful and deadly tropical cyclone that caused widespread damage in Eastern India and Bangladesh in May 2020. It made landfall near Bakkhali, West Bengal, on May 20, 2020, affecting both West Bengal and Odisha significantly.
C) Amphan — Super Cyclonic Storm Amphan hit Odisha and West Bengal in May 2020, causing extensive damage. It was one of the strongest cyclones to hit the region in decades.
To find which number divides the given sum, we can factor out the common term from all the addends. The smallest power of 3 in the expression is \(3^{71}\). Factoring this out will simplify the expression, making it easier to identify a divisor.
We know that \(121 = 11 \times 11 = 11^2\).
Since the expression can be written as a product that includes \(11^2\), it is clearly divisible by 11.
Correct Option: C) 11
This problem involves calculating means and sums of numbers. We are given the mean of six numbers and the mean of three of those numbers. We need to find the mean of the remaining three numbers. The key is to use the formula for the mean: \[\text{Mean} = \frac{\text{Sum of numbers}}{\text{Count of numbers}}\] to find the sums, and then deduce the required mean.
Calculate the sum of all six numbers:
The mean of a, b, c, d, e, and f is 36. There are 6 numbers.
Calculate the sum of b, d, and f:
The mean of b, d, and f is 28. There are 3 numbers.
Calculate the sum of a, c, and e:
The sum of all six numbers is the sum of (a, c, e) plus the sum of (b, d, f).
Calculate the mean of a, c, and e:
We have the sum of a, c, and e, which is 132. There are 3 numbers.
Thus, the mean of a, c, and e is 44.
Correct Option: C) 44
The question asks to identify the Indian state with the highest wind energy production as of 2021. This requires knowledge of the current status of renewable energy, specifically wind power, in various Indian states.
Correct Option: C) Tamil Nadu has the highest wind energy production in India as of 2021. It has consistently been the leading state in terms of installed wind power capacity and generation.
To find the momentum of an object, we use the fundamental formula that relates mass and velocity. Momentum is a vector quantity defined as the product of an object's mass and its velocity. The units must be consistent for the calculation.
Momentum (\(p\)) is given by the product of mass (\(m\)) and velocity (\(v\)).
\[ p = m \times v \]The unit of momentum is the product of the unit of mass (kg) and the unit of velocity (m/s), which is kg-m/s.
Correct Option: A) 392 kg-m/s is the calculated momentum.
The question asks to identify the Indian female shooter who won the gold medal in the individual 10m air rifle women's event at the 2019 ISSF World Cup in New Delhi. This requires knowledge of specific sports achievements and events.
Correct Option: A) Apurvi Chandela won the gold medal in the individual 10m air rifle women's event at the 2019 ISSF World Cup in New Delhi. She achieved a world record score of 252.9 in the final.
Study the given pattern carefully and select the number that can replace the question mark (?) in it.
| 9 | 5 | 5 |
| 5 | 7 | ? |
| 3 | 4 | 5 |
| 135 | 140 | 150 |