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What is the largest number which divides both \(2^{35}-1\) and \(2^{91}-1\) ?

Consider the following numbers :
1. 437
2. 797
3. 1073

How many of the above numbers are prime?

A, B, C, D and E enter into a business. They invest money in the ratio \( 2: 3: 4: 5: 6 \). However, the time invested by them is in the ratio \( 6: 5: 4: 3: 2 \). If the profit distributed is directly proportional to time and money invested, then who receives the highest amount of profit?

In a party of 150 persons, 75 persons take tea, 60 persons take coffee and 50 persons take milk. 15 of them take both tea and coffee, but no one taking milk takes tea. If each person in the party takes at least one drink, then what is the number of persons taking milk only?

What is the largest power of 10 that divides the product \(29 \times 28 \times 27 \times \ldots 2 \times 1\)?

What is the remainder when \(65^{99}\) is divided by 11?

If the roots of the equation \( x^{2}-b x+c=5 \) differ by 5, then which one of the following is correct?

Consider the following statements :
1. \( n^{3}-n \) is divisible by 6 .
2. \( n^{5}-n \) is divisible by 5 .
3. \( n^{5}-5 n^{3}+4 n \) is divisible by 120 .

Which of the statements given above are correct?

A can do a certain work at twice the speed of B. Further, B can do the same work at 1.5 times the speed of C. All of them together can finish the work in 12 days. In how many days can C alone finish the work?

The sum of digits of a 2-digit number is 12. When the digits are reversed, the number becomes greater by eighteen. What is the difference between the digits in the number?

The time taken by a train to cross a man travelling in another train is 10 seconds, when the other train is travelling in the opposite direction. However, it takes 20 seconds, if both the trains are travelling in the same direction. The length of the first train is 200 m and that of the second train is 150 m. What is the speed of the first train?

If a, b, c, d, e and f satisfy \(2a = 3b = 6c = 9d = 12e = 18f\), then what is the value of \((a+b)/(c+d+e+f)\)?

If \( a, b, c \) are non-zero real numbers such that \( a+b+c=0 \), then what are the roots of the equation \( ax^2 + bx + c = 0 \) ?

Twelve percent of bananas bought by a fruit vendor got lost during transportation. On selling the remaining bananas, the vendor's overall profit turned out to be \(4\%\). If the vendor had not lost any bananas and had sold them at the price of the remaining bananas, what would have been his profit percentage?

If the positive square root of \( (5+3 \sqrt{2})(5-3 \sqrt{2}) \) is \( \alpha \), then what is the positive square root of \( 8+2 \alpha \)?

When every even power of every odd integer (greater than 1) is divided by 8, what is the remainder?

Consider the following statements:
1. If \(n\) is a natural number, then the number \(\frac{n(n^2+2)}{3}\) is also a natural number.
2. If \(m\) is an odd integer, then the number \(\frac{m^4 + 4m^2 + 11}{16}\) is an integer.
Which of the statements given above is/are correct?

It is given that 5 does not divide \(n-1\), \(n\) and \(n+1\), where n is a positive integer. Which one of the following is correct?

What is the largest 5 -digit number, which leaves remainder 7 , when divided by 18 as well as by 11 ?

In a business dealing, \(A\) owes \(B\) ₹20,000 payable after 5 years, whereas \(B\) owes \(A\) ₹12,000 payable after 4 years. They want to settle it now at the rate of \(5\%\) simple interest. Who gives how much money in this settlement?

Average marks in Mathematics of Section A comprising 30 students is 65 and that of Section B comprising 35 students is 70. What are the average marks (approximately) of both the sections if it was detected later that an entry of 47 marks was wrongly made as 74?

If \( \alpha \) and \( \beta \) are the roots of the equation \( x^{2}-7x+1=0 \), then what is the value of \( \alpha^{4}+\beta^{4} \)?

Consider the following statements in respect of all factors of 360:
1. The number of factors is 24.
2. The sum of all factors is \(1170\).
Which of the above statements is/are correct?

Consider a 6-digit number of the form XYXYXY. The number is divisible by :

What is the HCF of \( 3^{29}-9 \) and \( 3^{38}-9 \) ?

If \( x=\sqrt{4 \sqrt{4 \sqrt{4 \sqrt{4 \ldots}}}} \), then what is the value of \( x \)?

Let \( m \) and \( n \) be natural numbers. What is the minimum value of \(m+n\) such that \(33m + 22n\) is divisible by \(121\)?

The product of two numbers is 2160 and their HCF is 12. If the sum of the squares of the two numbers is 4896, then what is the mean of the two numbers?

The age of \( Q \) exceeds the age of \( P \) by 3 years. The age of \( R \) is twice the age of \( P \) and the age of Q is twice the age of S . Further, the age difference of \( R \) and \( S \) is 30 years. What is the sum of the ages of P and Q ?

If \( a, b \) and c are the sides of a triangle ABC, then \( \sqrt{a} + \sqrt{b} - \sqrt{c} \) is always:

There are four bells which ring at an interval of 15 minutes, 25 minutes, 35 minutes and 45 minutes respectively. If all of them ring at 9 A.M., how many more times will they ring together in the next 72 hours?

Let a, b, c and d be four positive integers such that \(a+b+c+d=200\). If \(S=(-1)^a+(-1)^b+(-1)^c+(-1)^d\), then what is the number of possible values of \(S\)?

The number \( 97^{30}-14^{30} \) is divisible by :

Consider the following statements :
1. \( \log_{10} 50 \) is a rational number.
2. \( \log_{100} 10 \) is an irrational number.

Which of the statements given above is/are correct?

If 17 women and 24 men can do a piece of work in 5 days and 12 women and 23 men can do it in 6 days, then which one of the following is correct?

Three taps A, B and C together can fill a tank in 6 hours. Tap C alone can fill the tank in 12 hours. To fill the tank, when it is empty, all the three taps are started together. After working t hours, tap C is closed and the tank is filled in 8 more hours. What is t equal to?

A, B and C can complete a work in x, 1.5x and 2 x days respectively. If they complete the work together, in what ratio should they be paid?

What is the last digit of the sum \( \mathrm{S}=9^{27}+27^{9} \)?

If \( x=\frac{\sqrt{3}+1}{\sqrt{3}-1} \) and \( y=\frac{\sqrt{3}-1}{\sqrt{3}+1} \), then what is the value \( x^{3}-y^{3} \)?

The speed of a boat in still water is \(15\ \mathrm{km}/\mathrm{hr}\). If it can travel 42 km downstream and 28 km upstream in the same time, then what is the speed of the stream?