Number system: practice set 1

X, Y and Z jump forward 4′, 6′ and 5′, respectively. At 8 AM, they all land on mark 199′. How many times will they all land on the same mark (need not be at the same moment) between mark 195′ and 1000′, if all of them cross mark 1000′ by 9 AM?

How many times does 5 appear in all two-digit positive integers?

How many possible values of (p + q + r) are there satisfying 1/p + 1/q + 1/r = 1, where p, q and r are natural numbers (not necessarily distinct)?

Let PQR be a 3-digit number, PPT be a 3-digit number and PS be a 2-digit number, where P, Q, R, S, T are distinct non-zero digits. Further, PQR − PS = PPT. If Q = 3 and T < 6, then what is the number of possible values of (R, S)?

Consider the following statements:

1. I.There exists a natural number which when increased by 50% can have its number of factors unchanged.
2. II.There exists a natural number which when increased by 150% can have its number of factors unchanged.

Which of the statements given above is/are correct?

On January 1st, 2023, a person saved ₹1. On January 2nd, 2023, he saved ₹2 more than that on the previous day. On January 3rd, 2023, he saved ₹2 more than that on the previous day and so on. At the end of which date was his total savings a perfect square as well a perfect cube?

Let p, q, r and s be distinct positive integers. Let p, q be odd and r, s be even. Consider the following statements:

1. 1.(p − r)²(qs) is even.
2. 2.(q − s)q²s is even.
3. 3.(q + r)²(p + s) is odd.

Which of the statements given above are correct?

What is the remainder when 85 × 87 × 89 × 91 × 95 × 96 is divided by 100?

For any choices of values of X, Y and Z, the 6-digit number of the form XYZXYZ is divisible by:

How many natural numbers are there which give a remainder of 31 when 1186 is divided by these natural numbers?

What is the value of X in the sequence 20, 10, 10, 15, 30, 75, X?

What is the smallest number greater than 1000 that when divided by any one of the numbers 6, 9, 12, 15, 18 leaves a remainder of 3?

The number 3798125P369 is divisible by 7. What is the value of the digit P?

The difference between a 2-digit number and the number obtained by interchanging the positions of the digits is 54.

Consider the following statements:

1. 1.The sum of the two digits of the number can be determined only if the product of the two digits is known.
2. 2.The difference between the two digits of the number can be determined.

Which of the above statements is/are correct?

If you have two straight sticks of length 7.5 feet and 3.25 feet, what is the minimum length can you measure?

The recurring decimal representation 1.272727... is equivalent to

The number of times the digit 5 will appear while writing the integers from 1 to 1000 is

What is X in the sequence 4, 196, 16, 144, 36, 100, 64, X?

If X is between −3 and −1, and Y is between −1 and 1, then X² − Y² is in between which of the following?

The age of Mr. X last year was the square of a number and it would be the cube of a number next year. What is the least number of years he must wait for his age to become the cube of a number again?