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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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.
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\(14927 \times 567 - 14927\times 467 = y\) . What is \(y\) ?
Answer: B
\(y = 14927 \times 567 - 14927 \times 467\)
\(= 14927 \times (567 - 467)\)
\(= 14927 \times 100\)
\(= 1492700\)
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If a person makes a row of toys of 20 each, there would be 15 toys left.
If they made to stand in rows of 25 each, there would be 20 toys left, if they made to stand in rows of 38 each, there would be 33 toys left and if they are made to stand in rows of 40 each, there would be 35 toys left.
The minimum number of toys the person has is?
Answer: D
Required number of toys =LCM(20,25,28,38 and 40)–5=LCM(20,25,28,38 and 40)–5
⇒ 3,800−5=3,800−5= 3,795.
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How many keystrokes are needed to type numbers from 1 to 1000 on a standard keyboard?
Answer: B
While typing numbers from 1 to 1000, you have 9 single digit numbers from 1 to 9. Each of them requires one keystroke. That is 9 key strokes.
There are 90 two-digit numbers, from 10 to 99. Each of these numbers requires 2 keystrokes. Therefore, one requires 180 keystrokes to type the 2 digit numbers.
There are 900 three-digit numbers, from 100 to 999. Each of these numbers requires 3 keystrokes. Therefore, one requires 2700 keystrokes to type these 3 digit numbers.
Then 1000 is a four-digit number which requires 4 keystrokes.
Totally, therefore, one requires 9+180+2700+4=9+180+2700+4= 2893 keystrokes.
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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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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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The sum of the two digits of a number is 10. If the number is subtracted from the number obtained by reversing its digits, the result is 54. Find the number?
Answer: B
Any two digit number can be written as \((10P + Q)\), where \(P\) is the digit in the tens place and \(Q\) is the digit in the units place.
\(P + Q = 10\) ----- (1)
\((10Q + P) - (10P + Q) = 54\)
\(9(Q - P) = 54\)
\((Q - P) = 6\) ----- (2)
Solve (1) and (2) \(P = 2\) and \(Q = 8\)
The required number is \(= 28 \)
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How many ordered pairs of integer (x,y)(x,y) are there such that their product is a positive integer less than 100.
Answer: D
Given 0
Also given (x,y) is not equal to (y,x).
If x=1, y can take values from 1 to 99
⇒ we have 99×2=198 pairs but (1,1) is repeated
Thus can take 198−1=197 pairs.
If x=2 , yy can take values from 2 to 49
[(2,1) and (1,2) are also covered in 197 pairs above].
⇒ 48×2−1=95 pairs [(2,2) is repeated]
Similarly,
If x=3 or y=3 we have 61 pairs
If x=4 or y=4 we have 41 pairs
If x=5 or y=5 we have 29 pairs
If x=6 or y=6 we have 21 pairs
If x=7 or y=7 we have 15 pairs
If x=8 or y=8 we have 9 pairs
If x=9 or y=9 we have 5 pairs
We have total 473 pairs when x and y are positive.
We will have 473 pairs when a and b are negative.
⇒ We have a total of 946 ordered pairs.
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When 75% of a two-digit number is added to it, the digits of the number are reversed. Find the ratio of the unit's digit to the ten's digit in the original number.
Answer: C
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