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# Pre RMO Sample Papers With Answer Keys

In this post, some important questions of Pre RMO Sample Papers With Answer Keys are given.

Pre RMO Sample Papers with Answer Keys Question No 1:

When a three digit number is divided by 2, 3, 4, 5 and 7, the remainders are all 1. Find the minimum and maximum values of such three digit numbers.

Solution:

Pre RMO Sample Papers with Answer Keys Question No 2:

Find the remainder of the number

when divided by 7

Solution:

Pre RMO Sample Question No 3:

If (x –1) (x + 3) (x –4) (x – 8) + m is a perfect square, then m is?

Solution:

Pre RMO Sample Question No 4:

If n + 20 and n – 21 are both perfect squares, where n is a natural number, find n.

Solution:

Pre RMO Sample Question No 5:

If a four digit number and the sum of its all digits have a sum 2006, find the four digit number

Solution:

Pre RMO Sample Question No 6:

Given that the equation 4x/3 – a = 2x/5 + 140 has a positive integer solution, where a is a parameter. Find the minimum positive integer value of a.

Solution:

Pre RMO Sample Question No 7:

Find the positive integer solutions to the equation 12x + 5y = 125.

Solution:

Read More : Pre RMO 2019 Results & Cut Off Marks

Pre RMO Sample Question No 8:

Find the general solution of Diophantine equation 71x + 83y = 5

Solution:

Pre RMO Sample Question No 9:

A four digit number has remainder 13 when divided by 16, and has remainder 122 when divided by 125. Find the minimum value of such four digit numbers.

Solution:

Pre RMO Sample Question No 10:

Determine if there is a natural number k such that the sum of the two numbers 3k2 + k – 4 and 7k2 – 3k+ 1 is a perfect square.

Solution:

Pre RMO Sample Question No 11:

Let S = , then find the value of .

Solution:

Pre RMO Sample Question No 12:

If the sum of 2009 consecutive positive integers is a perfect square, find the minimum value of the maximum number of the 2009 numbers.

Solution:

Pre RMO Sample Question No 13:

Solve the equation 2[x] = x + 2{x}.

Solution:

Pre RMO Sample Question No 14:

The digits 1, 2, 3, 4, and 5 are each used once to compose a five-digit number abcde such that the
three digit number abc is divisible by 4, bcd divisible by 5, and cde is divisible by 3. Find the digit a.

Solution:

Pre RMO Sample Question No 15:

Let abcdef be a 6-digit integer such that defabc is 6 times the value of abcdef. Find the value of a +b + c + d + e + f.

Solution:

Pre RMO Sample Question No 16:

Find all the natural number n such that n2 – 19n + 91 is a perfect square.

Solution:

Pre RMO Sample Question No 17:

What is the smallest positive integer n > 1 such that 3n ends with 003?

Solution:

Pre RMO Sample Question No 18:

Find all the positive integers n, such that n × 2n + 1 is divisible by 3.

Solution:

Pre RMO Sample Question No 19:

Given that five digit number 2x9y1 is a perfect square number. Find the value of 3x + 7y..

Solution:

Pre RMO Sample Question No 20:

Find the remainder of (25733 + 46)26 when it is divided by 50.

Solution:

Pre RMO Sample Question No 21:

Find the smallest positive integer n such that 1000 <= n <= 1100 and 1111n + 1222n + 1333n + 1444n is divisible by 10.

Solution:

Pre RMO Sample Question No 22:

Simplify M = .

Solution:

Pre RMO Sample Question No 23:

Simplify .

Solution:

Pre RMO Sample Question No 24:

Simplify .

Solution:

Pre RMO Sample Question No 25:

Find the last two digits of .

Solution:

Pre RMO Sample Question No 26:

What is the remainder of 1234567894 when it is divided by 8?

Solution:

Pre RMO Sample Question No 27:

Simplify

Solution:

Pre RMO Sample Question No 28:

Find the number of positive integer n, such that the remainder is 7 when 2007 is divided by n.

Solution:

Pre RMO Sample Question No 29:

Find the remainder of  when it is divided by 11.

Solution:

Pre RMO Sample Question No 30:

It is known that 2726, 4472, 5054, 6412 have the same remainder when they are divided by some two digit natural number m. Find the value of m

Solution:

1__ Min = 421 and Max = 841

2__ 6

3__ 196

4__ 421

5__ 1984

6__ Min Value of a = 14

7__ (5, 13) and (10, 1)

8__ x = -29 + 83t, y = 6 – 17t

9__ 1997

10__ Does Not Exist

11__ 241

12__ 1045

13__ x = 0, 4/3, 8/3

14__ 1

15__ 27

16__ 9 or 10

17__ 101

18__ 6k + 1 or 6k + 2, k = 0, 1, 2…

19__ 29

20__ 29

21__ 1001

22__

23__

24__ 36

25__ 88

26__ 1

27__

28__ 16

29__ 9

30__ 97

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