Mathematiq'19 Pre O Qualifying Round

Explanation
The HCF (Highest Common Factor) is the largest number that divides all the given numbers without leaving a remainder. To find the HCF of 160, 240, 480, and 360, we can factorize each number and find the common factors. The prime factorization of these numbers is: 160 = 2^5 * 5, 240 = 2^4 * 3 * 5, 480 = 2^5 * 3 * 5, and 360 = 2^3 * 3^2 * 5. The common factors are 2^3 * 5 = 40. Therefore, the HCF of the given numbers is 40.

Explanation
When rounding off a number to two decimal places, we look at the third decimal place. If it is 5 or greater, we round up the second decimal place. In this case, the third decimal place is 7, which is greater than 5. Therefore, we round up the second decimal place, resulting in 6.49.

Explanation
A number is divisible by 3 if the sum of its digits is divisible by 3. In the given number, 443262, the sum of the digits is 4 + 4 + 3 + 2 + 6 + 2 = 21, which is divisible by 3. Therefore, 443262 is divisible by 3.

Explanation
The expression (-4)3 means to raise -4 to the power of 3. When a negative number is raised to an odd power, the result is always negative. Therefore, (-4)3 is equal to -4 multiplied by -4 multiplied by -4, which equals -64.

Explanation
The number 64 is both a square number and a cube number because it can be expressed as 8^2 and 4^3. A square number is a number that can be obtained by multiplying an integer by itself, while a cube number is a number that can be obtained by multiplying an integer by itself twice.

Explanation
The price of one book is Rs. 30. A dozen is equal to 12, so the price of one dozen of such books would be 12 multiplied by Rs. 30, which equals Rs. 360.

Explanation
The LCM (Least Common Multiple) is the smallest multiple that is divisible by all the given numbers. To find the LCM of 18, 24, and 27, we can list the multiples of each number and find the smallest common multiple. The multiples of 18 are 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, 216... The multiples of 24 are 24, 48, 72, 96, 120, 144, 168, 192, 216... The multiples of 27 are 27, 54, 81, 108, 135, 162, 189, 216... The smallest common multiple is 216, so the LCM of 18, 24, and 27 is 216.

Explanation
The given expression is a sum of two fractions. To add fractions, we need to have a common denominator. In this case, the common denominator is 45. To convert 3/9 to a fraction with a denominator of 45, we multiply the numerator and denominator by 5. This gives us 15/45. Similarly, to convert 4/15 to a fraction with a denominator of 45, we multiply the numerator and denominator by 3. This gives us 12/45. Adding 15/45 and 12/45 gives us a numerator of 27 and a denominator of 45. Therefore, the correct answer is 27/45.

Explanation
The operation 3 - 1 will be evaluated first to simplify the expression because subtraction has a higher precedence than addition, multiplication, and division.

Explanation
The given expression can be simplified as follows: -2 + 8 + (-24 ÷ -2).

First, we solve the division part: -24 ÷ -2 = 12.

Then, we substitute the division result back into the expression: -2 + 8 + 12.

Next, we perform addition from left to right: -2 + 8 = 6, and 6 + 12 = 18.

Therefore, the correct answer is 18.

Explanation
The given expression involves addition and subtraction of numbers. Starting from the innermost parentheses, we add -8 and -6, which gives us -14. Then, we add 40 and -14, resulting in 26. Finally, we add -68, 26, and 35, which gives us -7. Therefore, the correct answer is -7.

Explanation
The fraction of a day that is 8 hours is 1/3 because there are 24 hours in a day. Dividing 8 by 24 gives us 1/3.

Explanation
The largest perfect square between 4 and 50 is 49. A perfect square is a number that can be expressed as the product of two equal integers. In this case, 49 is equal to 7 multiplied by 7, making it a perfect square. The other options, 25 and 36, are also perfect squares, but they are smaller than 49. 50, on the other hand, is not a perfect square as it cannot be expressed as the product of two equal integers. Therefore, the correct answer is 49.

Explanation
The numbers 52 and 78 both have 13 as a common factor because 13 can evenly divide both numbers.

Explanation
The question asks for the fraction of prime numbers among the numbers from 2 to 12. To find this fraction, we need to determine the number of prime numbers in that range. The prime numbers between 2 and 12 are 2, 3, 5, 7, 11. There are a total of 11 numbers from 2 to 12. Therefore, the fraction of prime numbers is 5/11.