The difference between a number and its three-fifths is 50. What is the number?
Answer: C
Three-fifths taken away leaves two-fifths.
\[ x - \frac{3x}{5} = \frac{2x}{5} = 50 \]
\[ x = 125 \]
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When a number is divided by 13, the remainder is 11. When the same number is divided by 17, the remainder is 9. What is the number?
Answer: B
Let the number be N.
N = 13k + 11
N = 17m + 9
We can test the options.
A) 339: \(339 \div 13 = 26\) rem 1. Incorrect.
B) 349: \(349 \div 13 = 26\) rem 11. Correct. Now check with 17. \(349 \div 17 = 20\) rem 9. Correct. So 349 is the answer.
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Find the value of \(\frac{(67895 \times 67895 - 37895 \times 37895)}{30000}\).
Answer: A
The numerator is in the form \(a^2 - b^2 = (a-b)(a+b)\).
Here a = 67895 and b = 37895.
\(a-b = 67895 - 37895 = 30000\).
\(a+b = 67895 + 37895 = 105790\).
The expression becomes \(\frac{(30000)(105790)}{30000} = 105790\).
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What is the average of the first 15 odd numbers?
Answer: A
The first 15 odd numbers sum to \( 15^{2} = 225 \).
\[ \text{average} = \frac{225}{15} = 15 \]
The average of the first \( n \) odd numbers is always \( n \) - they are symmetric about it.
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What is the smallest 6-digit number that is exactly divisible by 111?
Answer: D
The smallest 6-digit number is 100,000.
To find the smallest 6-digit number divisible by 111, we divide 100,000 by 111.
\(100000 \div 111\) gives a quotient of 900 and a remainder of 100.
The number to be added to 100,000 to make it divisible by 111 is \((111 - \text{remainder}) = 111 - 100 = 11\).
So, the required number is \(100,000 + 11 = 100,011\).
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The difference of two numbers is 1365. On dividing the larger number by the smaller, we get 6 as quotient and 15 as remainder. What is the smaller number?
Answer: B
Let the larger number be L and the smaller number be S.
Given: L - S = 1365 (Eq. 1)
From the division algorithm, L = 6S + 15 (Eq. 2)
Substitute L from Eq. 2 into Eq. 1:
(6S + 15) - S = 1365
5S + 15 = 1365
5S = 1350
S = \(1350 \div 5 = 270\).
The smaller number is 270.
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3/4 of a number is 60 more than its 1/3. The number is:
Answer: B
Let the number be x.
The equation is \(\frac{3}{4}x = \frac{1}{3}x + 60\).
\(\frac{3}{4}x - \frac{1}{3}x = 60\).
The LCM of 4 and 3 is 12.
\(\frac{9x - 4x}{12} = 60\).
\(\frac{5x}{12} = 60\).
\(x = 60 \times \frac{12}{5} = 12 \times 12 = 144\).
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The number 1 is:
Answer: C
A prime number is a natural number greater than 1 with exactly two distinct positive divisors: 1 and itself. Since 1 only has one divisor (1), it is not prime.
A composite number is a natural number that has at least one divisor other than 1 and itself. Since 1 does not fit this definition, it is not composite.
Therefore, 1 is a special case and is considered neither prime nor composite.
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Which digit should come in place of * in the number 8*9632 to make it divisible by 11?
Answer: A
A number is divisible by 11 if the difference of the sums of alternating digits is 0 or a multiple of 11.
Let's sum the digits in odd positions (from the right): 2 + 6 + * = 8 + *.
Let's sum the digits in even positions (from the right): 3 + 9 + 8 = 20.
The difference is \(|(8 + *) - 20| = |* - 12|\).
We need \(|* - 12|\) to be 0 or a multiple of 11.
If \(* - 12 = -11\), then \(* = 1\). Since * must be a single digit, this is a valid solution.
If \(* - 12 = 0\), * would be 12 (not a digit). If \(* - 12 = 11\), * would be 23 (not a digit).
So, the required digit is 1.
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\(? \div 25 \div 12 = 248.64\)
Answer: A
Undo the two divisions by multiplying.
\[ ? = 248.64 \times 12 \times 25 = 248.64 \times 300 \]
\[ = 74{,}592 \]
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