91.
\((935421 \times 625) = ?\)
Answer: B
Write 625 as \( \tfrac{10^{4}}{16} \).
\[ 935421 \times 625 = \frac{9{,}354{,}210{,}000}{16} = 584{,}638{,}125 \]
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Number System Questions and Answers
Important Instructions
92.
The largest 4 digit number exactly divisible by 88 is:
Answer: A
Divide the largest four-digit number by 88 and drop the remainder.
\[ 9999 = 88 \times 113 + 55 \]
\[ 9999 - 55 = 9944 = 88 \times 113 \]
9988 on the list is even and looks plausible, but \( 9988 \div 88 = 113.5 \).
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93.
The unit digit in \(7^{153}\) is?
Answer: C
\(7^{153} = \left ( 7^{4}\right )^{38}\times 7\)
Now, unit digit of \(\left ( 7^{4}\right )^{38} = 1\)
Therefore, unit digit of \(7153 = 1 \times 7\)
\(= 7\)
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94.
x divides \(\left ( \left ( 49 \right )^{15} -1\right )\) completely. What is \(x\) ?
Answer: A
\(\left (x^{n}-1 \right )\) is divisible by \(x+1\) if \(x\) is an even number.
\(\left ( \left ( 49 \right )^{15} -1\right )\) =\(\left ( \left ( 7^{2} \right )^{15} -1\right )\) =\(\left ( 7 \right )^{30} -1\)
which is divisible by \(7+1\), \(8\)
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95.
\(14927 \times 567 - 14927\times 467 = y\) . What is \(y\) ?
Answer: B
\(y = 14927 \times 567 - 14927 \times 467\)
\(= 14927 \times (567 - 467)\)
\(= 14927 \times 100\)
\(= 1492700\)
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96.
\(9548 + 7314 = 8362 + x\). What is \(x\) ?
Answer: C
\[ x = 9548 + 7314 - 8362 \]
\[ = 16862 - 8362 = 8500 \]
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97.
What is the unit digit of \(13^{35}\) ?
Answer: D
\(13^{1}\) has unit digit \(3\)
\(13^{2}\) has unit digit \(9\)
\(13^{3}\) has unit digit \(7\)
\(13^{4}\) has unit digit \(1\)
This will form a pattern \(3-7-9-1\)
So, \(13^{32}\) has unit digit \(1\)
\(13^{35}\) has unit digit as \(7\)
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98.
The smallest prime number is?
Answer: C
A prime has exactly two distinct divisors, 1 and itself.
1 has only one divisor, and 0 has infinitely many, so neither qualifies. The smallest prime is 2 - and the only even one.
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99.
When a number is divided by 17, the remainder is 9. When the same number is divided by 23, the remainder is 4. Find the number?
Answer: B
\(x = 17p + 9\) and \(x = 23q + 4\)
i.e. \(17p + 9 = 23q + 4\)
Therefore, \(q = \frac{(17p + 5)}{23}\)
Least value of p for which q is a whole number is \(p = 20\)
\(x = 17p + 9\)
\(= 17 \times 20 + 9\)
\(= 349\)
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100.
\(\left ( 64 - 12 \right )^{2} + 4 × 64 × 12 = ?\)
Answer: D
Given statement is like \(\left ( a - b \right )^{2} + 4ab\) where \(a=64\) and \(b=12\)
\(\left ( a - b \right )^{2} + 4ab\)
\(=\left (a^{2} - 2ab+b^{2}\right )+4ab\)
\(=a^{2}+2ab+b^{2}\)
\(=\left ( a + b \right )^{2}\)
Hence,
\(\left ( 64 - 12 \right )^{2} + 4 × 64 × 12\)
\(=\left ( 64 + 12 \right )^{2}\)
\(=76^{2}\)
\(=5776\)
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