Class 6 Prime Time Solutions Ganita Prakash

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Chapter 5: Class 6 Prime Time Solutions

Class 6 Prime Time Solutions

5.1 Common Multiples and Common Factors

The Idli-Vada Game (Page 107)

This game introduces the ideas of multiples and common multiples.

  • ‘Idli’ is said for numbers that are multiples of 3 (3, 6, 9, 12, …).
  • ‘Vada’ is said for numbers that are multiples of 5 (5, 10, 15, 20, …).
  • ‘Idli-Vada’ is said for numbers that are multiples of BOTH 3 and 5.

Which numbers are multiples of both 3 and 5?

These numbers are called Common Multiples. To find them, we can find the multiples of the larger number (5) and check which ones are also divisible by 3.

  • 5 (No), 10 (No), 15 (Yes), 20 (No), 25 (No), 30 (Yes), 35 (No), 40 (No), 45 (Yes)…

The numbers are 15, 30, 45, and so on. These are the multiples of 15, which is the Least Common Multiple (LCM) of 3 and 5.

Figure it Out (Page 108)

1. At what number is ‘idli-vada’ said for the 10th time?

‘Idli-vada’ is said for common multiples of 3 and 5, which are the multiples of 15. The 10th such number is 10 × 15 = 150.

2. If the game is played for numbers 1 to 90, find out:

a. How many times ‘idli’? (Multiples of 3)
90 ÷ 3 = 30 times.

b. How many times ‘vada’? (Multiples of 5)
90 ÷ 5 = 18 times.

c. How many times ‘idli-vada’? (Multiples of 15)
90 ÷ 15 = 6 times.

3. What if the game was played till 900?

a. ‘Idli’: 900 ÷ 3 = 300 times.

b. ‘Vada’: 900 ÷ 5 = 180 times.

c. ‘Idli-vada’: 900 ÷ 15 = 60 times.

“Yesterday, we played this game… One of the numbers was 4.” The puzzle states they only said ‘idli’ or ‘idli-vada’, never just ‘vada’. This means every multiple of the ‘vada’ number is also a multiple of the ‘idli’ number (4).

We need to find a number from the list (2, 3, 5, 8, 10) whose multiples are all also multiples of 4.

  • Multiples of 2: 2, 4, 6, 8… (6 is not a multiple of 4). No.
  • Multiples of 8: 8, 16, 24, 32… (All are multiples of 4). Yes.

The other number could be 8. For example, at 8, you’d say ‘idli-vada’. At 12, you’d say ‘idli’. You’d never have a situation to say just ‘vada’.

Jump Jackpot (Page 109)

This game introduces the concept of Factors or Divisors. A jump size is a “factor” of the treasure number if a jump of that size lands exactly on the treasure.

  • Factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.
  • To land on two treasures (e.g., 14 and 36), the jump size must be a Common Factor of both numbers.
  • Factors of 14: {1, 2, 7, 14}. Factors of 36: {1, 2, 3, 4, 6, 9, 12, 18, 36}. Common Factors: {1, 2}.

Table Analysis & Figure it Out (Page 110-111)

1. Analysis of the number table (31-70):

  • Shaded numbers (33, 36, 39…) are all multiples of 3.
  • Circled numbers (32, 36, 40…) are all multiples of 4.
  • Both shaded and circled numbers (36, 48, 60) are common multiples of 3 and 4.

2. Who am I?

a. I am a multiple of 7, less than 40. The sum of my digits is 8.
Multiples of 7 < 40: 7, 14, 21, 28, 35.
Sum of digits for 35 is 3+5=8. The number is 35.

b. I am a multiple of 3 and 5 (so a multiple of 15), less than 100. One digit is 1 more than the other.
Multiples of 15 < 100: 15, 30, 45, 60, 75, 90.
Checking digits of 45: 4 and 5. The difference is 1. The number is 45.

3. Find a perfect number between 1 and 10.

A perfect number’s factors (including itself) sum to twice the number. Let’s check 6.

Factors of 6 are 1, 2, 3, 6.
Sum = 1 + 2 + 3 + 6 = 12.
Twice the number is 2 × 6 = 12.
Since the sums match, 6 is a perfect number.

8. In the diagram, the common multiples are 24, 48, 72. What could the two numbers be?

The common multiples (24, 48, 72) are all multiples of 24. This means the LCM of the two numbers is 24. There are many possibilities. A simple one is 6 and 8 (LCM is 24). Another is 3 and 8 (LCM is 24).
Let’s use 6 and 8.
Multiples of 6 (not 8): 6, 12, 18, 30, 36…
Multiples of 8 (not 6): 8, 16, 32, 40…

5.2 Prime and Composite Numbers

  • A Prime Number is a number greater than 1 that has exactly two factors: 1 and itself. (e.g., 2, 3, 5, 7, 11, 13…)
  • A Composite Number is a number greater than 1 that has more than two factors. (e.g., 4, 6, 8, 9, 10, 12…)
  • The number 1 is neither prime nor composite.

Sieve of Eratosthenes & Figure it Out (Page 113-114)

1. Is there any other even prime besides 2?

No. Every other even number is, by definition, divisible by 2. This means it has at least three factors (1, 2, and itself), so it must be composite. 2 is the only even prime.

6. Find pairs of prime numbers up to 100 with the same digits (like 13 and 31).

These pairs are called Emirps.
The pairs are: (13, 31), (17, 71), (37, 73), and (79, 97).

7. Find seven consecutive composite numbers between 1 and 100.

We need a “prime desert”. Looking at the Sieve, the largest gap is after 89.
The seven consecutive composite numbers are: 90, 91, 92, 93, 94, 95, 96.

8. Find the twin primes between 1 and 100.

Twin primes are pairs of primes with a difference of 2.
The pairs are: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73).

5.3 Co-prime Numbers

Two numbers are co-prime if their only common factor is 1. They don’t have to be prime themselves. For example, 8 and 9 are co-prime.

Interesting Fact: The LCM of two co-prime numbers is simply their product. (e.g., LCM of 8 and 9 is 72).

5.4 Prime Factorisation

Prime Factorisation is the process of breaking down a composite number into a product of its prime factors. The Fundamental Theorem of Arithmetic states that every composite number has a unique prime factorisation.

Example: 56 = 2 × 2 × 2 × 7

Figure it Out (Page 120)

1. Find the prime factorisations of: 64, 105, 1000.

  • 64 = 2 × 32 = 2 × 2 × 16 = … = 2 × 2 × 2 × 2 × 2 × 2
  • 105 = 5 × 21 = 3 × 5 × 7
  • 1000 = 10 × 100 = (2 × 5) × (10 × 10) = … = 2 × 2 × 2 × 5 × 5 × 5

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

The number is 2 × 3 × 3 × 11 = 2 × 9 × 11 = 18 × 11 = 198.

5.5 Divisibility Tests

A number is divisible by:
  • 2 if its last digit is even (0, 2, 4, 6, 8).
  • 5 if its last digit is 0 or 5.
  • 10 if its last digit is 0.
  • 4 if the number formed by its last two digits is divisible by 4.
  • 8 if the number formed by its last three digits is divisible by 8.

Figure it Out (Page 125)

2. Find the largest and smallest 4-digit numbers that are divisible by 4 and are also palindromes.

A 4-digit palindrome has the form abba. For the number to be divisible by 4, the number formed by its last two digits (ba) must be divisible by 4.

  • Smallest: We start with the smallest ‘a’, which is 1. The number is 1bb1. The last two digits form b1. No two-digit number ending in 1 is divisible by 4. So ‘a’ cannot be 1.
    Let’s try ‘a’ = 2. The number is 2bb2. The last two digits form b2. The smallest ‘b’ that makes b2 divisible by 4 is b=1 (for 12). So the number is 2112.
  • Largest: We start with the largest ‘a’, which is 9. The number is 9bb9. The last two digits form b9. No two-digit number ending in 9 is divisible by 4.
    Let’s try ‘a’ = 8. The number is 8bb8. The last two digits form b8. We want the largest ‘b’. If b=8, we get 88, which is divisible by 4. So the number is 8888.

SUMMARY

  • Factors divide a number exactly. Multiples are the result of multiplying a number.
  • Prime Numbers have exactly two factors (1 and self). Composite Numbers have more than two.
  • Co-prime Numbers have no common factors other than 1.
  • Prime Factorisation is writing a number as a unique product of its prime factors. It is a powerful tool to check for co-primality and divisibility.
  • Divisibility Tests are shortcuts to check if a number is a factor of another without performing long division.