\((112 \times 5^{4}) = ?\)
Answer: B
\[ 5^{4} = 625 \]
\[ 112 \times 625 = 112 \times \frac{10000}{16} = \frac{1{,}120{,}000}{16} = 70{,}000 \]
Writing 625 as \( \tfrac{10^{4}}{16} \) turns the multiplication into a division by 16.
Please log in or register to report this question.
It is being given that \((2^{32} + 1)\) is completely divisible by a whole number. Which of the following numbers is completely divisible by this number?
Answer: D
Let \(2^{32} = x\).
Then, \((2^{32} + 1) = (x + 1)\).
Let \((x + 1)\) be completely divisible by the natural number N. Then,
\((2^{96} + 1) \\= [(2^{32})^{3} + 1]\\= (x^{3} + 1)\\= (x + 1)(x^{2} - x + 1)\)
which is completely divisible by N, since \((x + 1)\) is divisible by N.
Please log in or register to report this question.
What least number must be added to 1056, so that the sum is completely divisible by 23 ?
Answer: A
Divide and look at the remainder.
\[ 1056 = 23 \times 45 + 21 \]
\[ 23 - 21 = 2 \]
So adding 2 gives \( 1058 = 23 \times 46 \). Adding the remainder itself would make it divisible only by coincidence - it is the shortfall to the next multiple that is wanted.
Please log in or register to report this question.
1397 x 1397 = ?
Answer: A
\(1397\times 1397 = (1397)^{2}\)
\(= (1400-3)^{2}\)
\(= (1400)^{2}+(3)^{2}-(2\times 1400\times 3)\)
\(= 1960009 - 8400\)
\(= 1951609\)
Please log in or register to report this question.
The difference between the local value and face value of 7 in the numeral 657903 is:
Answer: B
The local (place) value of a digit is the digit multiplied by its place; the face value is the digit itself.
In 657903 the 7 sits in the thousands place:
\[ 7000 - 7 = 6993 \]
Please log in or register to report this question.
A number when divided by the sum of 555 and 445 gives two times their difference as quotient and 30 as the remainder. The number is:
Answer: D
\[ \text{divisor} = 555 + 445 = 1000 \]
\[ \text{quotient} = 2(555 - 445) = 220 \]
\[ \text{number} = 1000 \times 220 + 30 = 220{,}030 \]
Please log in or register to report this question.
The sum of three prime numbers is 100. If one of them exceeds another by 36, then one of the numbers is:
Answer: D
\(x+(x+36)+y=100 \\ \Rightarrow 2x+y=64\)
Therefore \(y\) must be even prime, which is 2.
Therefore,
\(2x+2=64 \\ \Rightarrow x=31\)
Third prime number
\(= (x+36)\\= (31+36)\\= 67\)
Please log in or register to report this question.
The unit’s digit in the product \((3127)^{173}\) is:
Answer: C
Unit digit in \((3127)^{173}\) = Unit digit in 7173. Now, 74 gives unit digit 1
Therefore, \(7173= (74)43 \times 71\). Thus, 7173 gives unit digit 7
Please log in or register to report this question.
A number when divided by 3 leaves a remainder 1. When the quotient is divided by 2, it leaves a remainder 1. What will be the remainder when the number is divided by 6?
Answer: C
Let \(n=3q+1\) and let \(q= 2p+1\). Then, \(n= 3(2p+1)+1= 6p+4\)
Therefore, the number when divided by \(6\), we get remainder\(= 4\)
Please log in or register to report this question.
The sum of three consecutive integers is 102. Find the lowest of the three?
Answer: D
Three consecutive numbers can be taken as \((P - 1), P, (P + 1)\)
So, \((P - 1) + P + (P + 1) = 102\)
\(3P = 102 => P = 34\)
The lowest of the three \(= (P - 1) = 34 - 1 = 33\)
Please log in or register to report this question.
