Class 6 Maths Chapter 5: Prime Time Notes

Common multiples and common factors

In the idli-vada game, children count in turn but say ‘idli’ for multiples of 3, ‘vada’ for multiples of 5 and ‘idli-vada’ for numbers that are multiples of both. The ‘idli-vada’ numbers, 15, 30, 45, …, are the common multiples of 3 and 5; they are the multiples of 15, so the 10th one is 150.

In the Jump Jackpot game, Jumpy starts at 0 and jumps by a fixed size. He lands on 24 with jump sizes 1, 2, 3, 4, 6, 8, 12 and 24, the factors of 24. To land on two treasures, such as 14 and 36, the jump size must be a common factor of both: 1 or 2.

Prime and composite numbers

Guna can arrange 12 figs in rectangles in several ways (1 × 12, 2 × 6, 3 × 4), but Anshu can arrange 7 figs only as 1 × 7. A number with exactly two factors, 1 and itself, is a prime number; a number with more than two factors is a composite number. The number 1, with only one factor, is neither. The first primes are 2, 3, 5, 7, 11, 13, 17 and 19, and 2 is the only even prime. Read more in prime and composite numbers.

The Sieve of Eratosthenes, named after a Greek mathematician of about 2200 years ago, finds primes: cross out 1, circle 2 and cross out its multiples, circle 3 and cross out its multiples, and so on. The circled numbers are prime; there are 25 up to 100.

Co-prime numbers

In a new treasure game, a jump size of 1 is not allowed. Grumpy’s treasures are safe on 4 and 9, because no jump size other than 1 reaches both. Two numbers with no common factor other than 1 are co-prime: 4 and 9 are co-prime, but 15 and 39 share the factor 3. For co-prime numbers, the first common multiple is their product. In thread art, a thread reaches every peg only when the number of pegs and the thread-gap are co-prime. Read more in co-prime numbers.

Prime factorisation

To check whether 56 and 63 are co-prime, Anshu writes 56 = 14 × 4 and 63 = 21 × 3 and wrongly decides they are, but Guna writes 56 = 7 × 8 and 63 = 9 × 7 and finds the common factor 7. The sure method is to break each number into primes: 56 = 2 × 2 × 2 × 7 and 63 = 3 × 3 × 7. This is the prime factorisation. Every number greater than 1 has exactly one, apart from the order of the factors, which does not matter because we can multiply in any order. Read more in prime factorisation.

Prime factorisations answer two questions. Two numbers are co-prime if they share no prime factor, like 40 = 2 × 2 × 2 × 5 and 231 = 3 × 7 × 11. One number divides another if its prime factorisation is included in the other’s: 168 = 2 × 2 × 2 × 3 × 7 is divisible by 12 = 2 × 2 × 3, but 42 = 2 × 3 × 7 is not, because it has only one 2.

Divisibility tests

Long division is not always needed. A number is divisible by 10 if its last digit is 0, by 5 if it ends in 0 or 5, and by 2 if it ends in 0, 2, 4, 6 or 8. For 4, look at the last two digits, and for 8, at the last three, because 100 is a multiple of 4 and 1000 a multiple of 8. So 8560 is divisible by 2, 4, 5, 8 and 10. Read more in divisibility rules.

Fun with numbers

In a box of four numbers, each number can be special in its own way: in 9, 16, 25 and 43, for example, 9 is the only single-digit number and 43 the only prime. In the prime puzzle, you fill a grid with primes so that each row and column multiplies to the given number.

Answers to every question are in the NCERT solutions for Chapter 5.