Find the remainder when PP is divided by 9?
Answer: C
We need to find sum of S of the digits of PP.
Till 999 each of the digits from 1 to 9 occurs equal number of times.
Total digit of all the numbers from 1 to 999 will be multiple of ∑9=45.
Hence,
S=45k+(1+0+0+0)
⇒ Required remainder = 1.
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Which digits should come in place of @ and # if the number 62684@# is divisible by both 8 and 5 ?
Answer: A
Since the given number is divisible by 5, so 0 or 5 must come in place of #.
But, a number ending with 5 is never divisible by 8.
So, 0 will replace #.
Now, the number formed by the last three digits is 4@0, which becomes divisible by 8, if @ is replaced by 4.
Hence, digits in place of @ and # are 4 and 0 respectively.
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If each of the three nonzero numbers a, b and c is divisible by 3, then abc must be divisible by which by ?
Answer: B
Since each one of the three numbers aa, bb, and cc is divisible by 3, the numbers can be represented as 3p,3qand 3r respectively, where p,q and r are integers.
The product of the three numbers is 3p×3q×3r=27(pqr).
Since p,qp,q and rr are integers, pqrpqr is an integer and therefore abcabc is divisible by 27.
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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\(= 4\)
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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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The sum of the present ages of two persons A and B is 60. If the age of A is twice that of B, find the sum of their ages 5 years hence?
Answer: C
\(A + B = 60, A = 2B\)
\(2B + B = 60 => B = 20\) then \(A = 40\)
5 years, their ages will be 45 and 25
Sum of their ages \(= 45 + 25 = 70\)
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Which of the following is the minimum value of the sum of two integers whose product is 36?
Answer: D
List all possible factors xx and yy whose product is 36, and calculate the corresponding sum x+yx+y:
| xx | yy | x×yx×y | x+yx+y |
| 1 | 36 | 36 | 37 |
| 2 | 18 | 36 | 20 |
| 3 | 12 | 36 | 15 |
| 4 | 9 | 36 | 13 |
| 5 | 6 | 36 | 12 |
From the table, the minimum sum is 12
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A number when divided by 6 leaves a remainder 3. When the square of the same number is divided by 6, the remainder is:
Answer: C
Let \(x=6q+3\). Then,
\(x^{2} = (6q+3)^{2} \\= 36q^{2}+36q+9 \\= 6(6q^{2}+6q+1)+3\)
So, when \(2n\) is divided by \(4\), remainder \(=3\)
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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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When a number divided by 342 gives remainder 47. When the same number is divided by 19, what would be the remainder?
Answer: B
On dividing the given number by 342, let kk be the quotient and 47 as remainder.
Then,
number−342k+47=(19×18k+19×2+9)number−342k+47=(19×18k+19×2+9)
=19(18k+2)+9=19(18k+2)+9
⇒ The given number when divided by 19, gives 18k+218k+2 as quotient and 9 as remainder.
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