Number System Questions and Answers

Answer: B

Write 625 as \( \tfrac{10^{4}}{16} \).

\[ 935421 \times 625 = \frac{9{,}354{,}210{,}000}{16} = 584{,}638{,}125 \]

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 \( 9988 \div 88 = 113.5 \).

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 \(7153 = 1 \times 7\)
\(= 7\)

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 \(7+1\), \(8\)

Answer: B

\(y = 14927 \times 567 - 14927 \times 467\)
\(= 14927 \times (567 - 467)\)
\(= 14927 \times 100\)
\(= 1492700\)

Answer: C

\[ x = 9548 + 7314 - 8362 \]

\[ = 16862 - 8362 = 8500 \]

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 \(7\)

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.

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 \(p = 20\)
\(x = 17p + 9\)
\(= 17 \times 20 + 9\)
\(= 349\)

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,
\(\left ( 64 - 12 \right )^{2} + 4 × 64 × 12\)
\(=\left ( 64 + 12 \right )^{2}\)
\(=76^{2}\)
\(=5776\)