There are three kinds of marking on a rope. The red markings divide the rope evenly into 10 parts. The blue markings divide the rope equally into 12 parts. The yellow markings divide the rope evently into 15 parts. How many segments does the rope have now?

2000 soldiers line up in a row. In the first round . They report "1,2", "1,2" and so on. Those who report "2" remain in the line. From the 2nd round onwards, they report in this manner: 1,2,3,1,2,3,... In each round, those who report "2" remain in the line, until only one soldier is left. What is the original number of this soldier

The lowest common multiple of a and b is 50. List all possible values of a and b

The greatest common divisor ò m and n is 15 . Given 3m + 2n = 225,

find mn.

A cardboard 140cm x 240cm is cut into many congruent squares. What is the largest possible square size ? How many suqares can be cut from the cardboard without wastage?

The product of a grandfather's age and his grandchild's age is 1330 next year. How old are they now?

The remainder is 2 when n is divided by 3. The remainder is 3, 4 and 5 when the divisors are 4, 5 and 6 respectively. What is the smallest possible of n?

Tiles of 30 cm x 45 cm are used to from a square flooring . The minimum number of tiles used is y pieces Find y?

The greatest common divisor of a and b is 8 . Their lowest common multiple is 96

a+b = ? 

a * b * c * d  * e = 2000, where each of a,b,c,d and e is greater than 1. The smallest possible value of a + b + c +d + e is ? 

The greatest common divisor of (3432, 5460) is ?

The number 1998 has x factors . Find x

There are 71 cards numbered from 1,2,3,...,71 Show that these cards can be arranged in a line, so that the sum of any two neighbouring numbers is a prime number

A mathematical competition has 30 questions.

5 marks are awarded for a correctly answered question.

1 mark is awarded for an unanswered question

1 mark is deducted  for an incorrectly answered question

Show that the total score of all participants is an even number

The sum of 2²⁰⁰⁹ +  3²⁰⁰⁹ +  7²⁰⁰⁹ + 9^{2009 is an odd number. Investigate whether the statement is true}

A page of a book was torn. The sum of the remaining page numbers is 3030. How many pages does the book have? which page was removed ?

it is given a,b and c are 3 consecutive odd numbers. Also, b³ = 3375. Find the value of ac

A number sequence is such that the first two numbers are both 1, From the 3rd number onwards, each one is the sum of the previous two terms. How many numbers in the sequence are odd numbers in the first 1000 numbers?

Where a and b are both integers and n = 10a + b, show that 23 | n if 23 | (a+7b).

Find the least value of (a+b) in a square number (123a + 984b), if both a and b are positive integers