Prime Time

Find any three numbers that are multiples of 25 but not multiples of 50.

Anshu and his friends play the 'idli-vada' game with two numbers, which are both smaller than 10. The first time anybody says 'idli-vada' is after the number 50. What could the two numbers be which are assigned 'idli' and 'vada'?

Find three prime numbers, all less than 30, whose product is 1955.

Which of the following numbers are divisible by all of 2, 4, 5, 8 and 10: 572, 2352, 5600, 6000, 77622160.

If the game is played for the numbers from 1 till 90, find out: a. How many times would the children say 'idli' (including the times they say 'idli-vada')? b. How many times would the children say 'vada' (including the times they say 'idli-vada')? c. How many times would the children say 'idli-vada'?

Twin pr \times are pairs of pr \times having a difference of 2. For example, 3 and 5 are twin primes. So are 17 and 19. Find the other twin pr \times between 1 and 100.

Observe that 3 is a prime number, and $2 \times 3$ + 1 = 7 is also a prime. Are there other pr \times for which doubling and adding 1 gives another prime? Find at least five such examples.

What is the smallest number whose prime factorisation has: a. three different prime numbers? b. four different prime numbers?

Find the remainders obtained when each of the following numbers are divided by i) 10, ii) 5, iii) 2. 78, 99, 173, 572, 980, 1111, 2345

What if the game was played till 900? How would your answers change?

A number for which the sum of all its factors is equal to twice the number is called a perfect number. The number 28 is a perfect number. Its factors are 1, 2, 4, 7, 14 and 28. Their sum is 56 which is twice 28. Find a perfect number between 1 and 10.

Write three pairs of prime numbers less than 20 whose sum is a multiple of 5.

Which of the following numbers is the product of exactly three distinct prime numbers: 45, 60, 91, 105, 330?

Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer. a. 30 and 45 b. 57 and 85 c. 121 and 1331 d. 343 and 216

Explore and find out if each statement is always true, sometimes true or never true. You can give examples to support your reasoning. a. Sum of two even numbers gives a multiple of 4. b. Sum of two odd numbers gives a multiple of 4.

At what number is 'idli-vada' said for the 10th time?

Find all multiples of 40 that lie between 310 and 410.

Find the smallest number that is a multiple of all the numbers from 1 to 10 except for 7.

The prime factorisation of a number has one 2, two 3s, and one 11. What is the number?

Find seven consecutive composite numbers between 1 and 100.