Online Mock Test 1: Number System

Explanation
To find the smallest possible value of 'a', we need to find the common factors of 112 and 33. The prime factorization of 112 is 2^4 * 7, and the prime factorization of 33 is 3 * 11. The common factors are 1 and 11. Since 'a' needs to be a factor of the given number, it must be divisible by 11. Among the given options, 363 is the smallest number that is divisible by 11. Therefore, the smallest possible value of 'a' is 363.

Explanation
To find the greatest number of five digits that is divisible by 7, 10, 15, 21, and 28, we need to find the least common multiple (LCM) of these numbers. The LCM of 7, 10, 15, 21, and 28 is 420, which means that any multiple of 420 is divisible by all these numbers. To find the greatest number of five digits, we need to find the largest multiple of 420 that is less than 100,000. The largest multiple of 420 that is less than 100,000 is 99,960. Therefore, the correct answer is 99960.

Explanation
A number is divisible by 3 if the sum of its digits is divisible by 3.
10^25 - 7 can be written as 9999999999999999999999993.
The sum of the digits is 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 + 3 = 225.
225 is divisible by 3, so 10^25 - 7 is also divisible by 3.

Explanation
When a number is divided by 36 and leaves a remainder of 19, it means that the number can be expressed as 36n + 19, where n is an integer. To find the remainder when this number is divided by 12, we substitute the expression into the division: (36n + 19) ÷ 12. Simplifying this expression, we get 3n + 1 with a remainder of 7. Therefore, the correct answer is 7.

Explanation
The G.C.D (Greatest Common Divisor) is the highest power of each variable that divides all the given terms. In this case, the highest power of x that divides all the terms is x^2, the highest power of y is y^2, and the highest power of z is z^2. Therefore, the G.C.D of 12x^2y^3z^2, 18x^3y^2z^4, and 24xy^4z^3 is 1x^2y^2z^2.

Explanation
The given answer "z is even" cannot be true because it contradicts the statement that z is odd. Since x and y are odd and positive, their sum will always be even. Therefore, the sum of x and y cannot be odd. The only statement that cannot be true is "z is even."

Explanation
The value of M and N respectively is 6 and 4. This is because for a number to be divisible by 8, the last three digits must be divisible by 8. In this case, the last three digits are 458, which is divisible by 8. Additionally, for a number to be divisible by 11, the difference between the sum of the digits in odd positions and the sum of the digits in even positions must be divisible by 11. In this case, the sum of the digits in odd positions is 3+0+4+8=15, and the sum of the digits in even positions is 9+4+5=18. The difference between these sums is 18-15=3, which is divisible by 11. Therefore, M=6 and N=4.

Explanation
In order to find the number of arrangements, we need to find the factors of 48 that are greater than or equal to 3. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. However, since we need at least 2 rows, we can eliminate the factors 1 and 48. Additionally, the number of students per row cannot exceed 16, so we can eliminate the factors 24 and 48. Thus, the possible arrangements are 2, 3, 4, 6, 8, 12, and 16, giving us a total of 7 arrangements.

Explanation
The sum of the first 100 numbers can be calculated by using the formula for the sum of an arithmetic series, which is (n/2)(first term + last term). In this case, the first term is 1 and the last term is 100, so the sum is (100/2)(1 + 100) = 50(101) = 5050. Since 5050 is an even number, it is divisible by 2. However, it is not divisible by 4 or 8, as these numbers require the sum to be divisible by 2 twice and three times respectively. Therefore, the correct answer is 2 only.

Explanation
Anita initially multiplied 35 with another number to get a product. However, she mistakenly used 53 instead of 35. As a result, the product increased by 540. To find the new product, we need to subtract 540 from the incorrect product she obtained. So, the correct answer is 1590.