Simplify: \(18 - [6 - \{4 - (8 - 6 + 3)\}]\).
Answer: B
Using BODMAS rule:
Step 1: Innermost bracket \((8 - 6 + 3) = (2 + 3) = 5\).
Step 2: Curly bracket \(\{4 - 5\} = -1\).
Step 3: Square bracket \([6 - (-1)] = [6 + 1] = 7\).
Step 4: Final calculation \(18 - 7 = 11\).
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The number 481481 is exactly divisible by which of the following?
Answer: D
Any number of the form 'abcabc' is a multiple of 1001.
\(481481 = 481 \times 1001\).
The prime factorization of 1001 is \(7 \times 11 \times 13\).
Therefore, any number of the form 'abcabc' will be divisible by 7, 11, and 13.
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What is the remainder when \(2^{31}\) is divided by 5?
Answer: C
We can use the concept of cyclicity of remainders.
\(2^1 \div 5\) gives remainder 2.
\(2^2 \div 5\) gives remainder 4.
\(2^3 \div 5\) gives remainder 3.
\(2^4 \div 5\) gives remainder 1.
\(2^5 \div 5\) gives remainder 2.
The cycle of remainders is (2, 4, 3, 1), which has a length of 4.
To find the remainder of \(2^{31} \div 5\), we divide the power 31 by the cyclicity 4.
\(31 \div 4\) gives a remainder of 3.
The remainder will be the 3rd term in the cycle, which is 3.
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The sum of all two-digit numbers divisible by 5 is:
Answer: C
The two-digit numbers divisible by 5 are 10, 15, 20, ..., 95. This is an arithmetic progression.
First term (a) = 10. Last term (l) = 95. Common difference (d) = 5.
Number of terms (n) = \((\frac{l-a}{d}) + 1 = (\frac{95-10}{5}) + 1 = (\frac{85}{5}) + 1 = 17 + 1 = 18\).
Sum = \(\frac{n}{2}(a+l) = \frac{18}{2}(10+95) = 9 \times 105 = 945\).
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What is the result of \(3.5 \times (60 \div 2.5)\)?
Answer: A
Using BODMAS, we solve the bracket first.
\(60 \div 2.5 = 60 \div (\frac{5}{2}) = 60 \times \frac{2}{5} = 12 \times 2 = 24\).
Now, multiply by 3.5: \(3.5 \times 24 = (\frac{7}{2}) \times 24 = 7 \times 12 = 84\).
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If a number is decreased by 4 and divided by 6, the result is 8. What would be the result if 2 is subtracted from the number and then it is divided by 5?
Answer: B
Let the number be x.
According to the first condition: \(\frac{x-4}{6} = 8\).
\(x-4 = 48\), so \(x = 52\).
Now, according to the second condition, subtract 2 from the number: \(52 - 2 = 50\).
Then divide by 5: \(50 \div 5 = 10\).
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A boy was asked to multiply a number by 25. He instead multiplied the number by 52 and got the answer 324 more than the correct answer. What was the number?
Answer: A
The excess comes from the difference between the two multipliers.
\[ 52x - 25x = 324 \quad\Longrightarrow\quad 27x = 324 \]
\[ x = 12 \]
Checking: \( 52 \times 12 = 624 \) and \( 25 \times 12 = 300 \), a difference of 324.
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Which of the following numbers is divisible by 9?
Answer: B
A number is divisible by 9 if the sum of its digits is a multiple of 9.
A) 7+5+3+2+0 = 17 (Not divisible by 9)
B) 7+5+3+2+1 = 18 (Divisible by 9, since 18 = 9 × 2)
C) 7+5+3+2+2 = 19 (Not divisible by 9)
D) 7+5+3+2+3 = 20 (Not divisible by 9)
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Find the sum of first 30 natural numbers.
Answer: B
\[ S_{n} = \frac{n(n+1)}{2} = \frac{30 \times 31}{2} = 465 \]
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Simplify: \(1 \div [1 \div \{1 \div (1 \div 1)\}]\).
Answer: A
Work outward from the innermost bracket.
\[ 1 \div 1 = 1 \quad\Longrightarrow\quad 1 \div 1 = 1 \quad\Longrightarrow\quad 1 \div 1 = 1 \]
Every division returns 1, so the whole expression is 1.
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