Chapter 5 · Prime Time

Ganita Prakash · Grade 6 · Complete Notes

Table of Contents

5.1 Common Multiples and Common Factors

The Idli–Vada Game

Children sit in a circle and count out numbers. The rule for what to say is:

Turn falls on…	Player says
A multiple of 3	idli
A multiple of 5	vada
A multiple of both 3 and 5	idli-vada
Any other number	the number itself

Say “idli” for 3, 6, 9, 12, 15… and “vada” for 5, 10, 15, 20…

The first number where a player says idli-vada is 15 — it is a multiple of 3 and of 5. Numbers that are multiples of both 3 and 5 (like 15, 30, 45, 60…) are called common multiples of 3 and 5.

Common Multiple: A number that appears in the multiplication tables of two (or more) numbers.
Common Factor: A number that divides two (or more) numbers exactly (no remainder).

Venn Diagram (Multiples up to 30)

Jump Jackpot – Factors

Jumpy starts at 0 and jumps in equal steps. If treasure is at 24, jump sizes that land on it are 1, 2, 3, 4, 6, 8, 12, 24 — these all divide 24 exactly and are called factors (divisors) of 24.

For two treasures, the jump sizes that reach both are the common factors of the two numbers.

Example: Treasures on 14 and 36.
Factors of 14 → 1, 2, 7, 14 | Factors of 36 → 1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors → 1 and 2. So only jump sizes 1 or 2 reach both.

Figure it Out — Page 108

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

Ans: 150 (10th common multiple of 3 and 5 = 15 × 10).

2. Game played for 1 to 90:

a) ‘idli’ (multiples of 3) → 30 times
b) ‘vada’ (multiples of 5) → 18 times
c) ‘idli-vada’ (multiples of 15) → 6 times

3. What if played till 900?

Ans: ‘idli-vada’ = 60 times, ‘vada’ = 180 times, ‘idli’ = 300 times.

4. Is the Venn figure related to the game?

Ans: Yes. The overlapping (common) numbers are exactly where you say ‘idli-vada’.

Page 109 – Other number that gives only ‘idli’ or ‘idli-vada’ (never just ‘vada’), one number is 4:

Ans: 8 (multiples of 8 are all multiples of 4, so ‘vada’ is never said alone).

Page 110 – Jump size that reaches both 15 and 30:

Ans: 1, 3, 5, 15.

Table (shaded = multiples of 3, circled = multiples of 4):

1. Shaded numbers are all multiples of 3.
2. Circled numbers are all multiples of 4.
3. Both shaded and circled → 36, 48, 60 = common multiples of 3 and 4.

Figure it Out — Pages 110–111

1. Multiples of 40 between 310 and 410:

Ans: 320, 360, 400.

2. Who am I?

a) Less than 40, factor 7, digit-sum 8 → 35.
b) Less than 100, factors 3 and 5, one digit 1 more than the other → 45.

3. Perfect number between 1 and 10:

Ans: 6 (factors 1 + 2 + 3 + 6 = 12 = twice 6).

4. Common factors of:

a) 20 & 28 → 1, 2, 4
b) 35 & 50 → 1, 5
c) 4, 8 & 12 → 1, 2, 4
d) 5, 15 & 25 → 1, 5

5. Three multiples of 25 that are not multiples of 50:

Ans: 25, 75, 125.

6. Two numbers below 10 whose first ‘idli-vada’ comes after 50:

Ans: 7 and 8 (first common multiple = 56), or 8 and 9 (= 72).

7. Jump sizes landing on both 28 and 70:

Ans: 1, 2, 7, 14 (common factors of 28 and 70).

8. Venn diagram with commons 72, 48, 24 — find the two numbers:

Ans: The numbers are Multiples of 3 and Multiples of 4 (since 24, 48, 72 are common multiples of 3 and 4).

9. Smallest number that is a multiple of all of 1–10 except 7:

Ans: 360.

10. Smallest number that is a multiple of all of 1–10:

Ans: 2520.

5.2 Prime Numbers

When you try to arrange objects in a rectangle, the number of rows and columns are its factors. 12 figs can make several rectangles (1×12, 2×6, 3×4), but 7 figs make only one (1×7).

Prime number: a number with exactly two factors — 1 and itself. First few: 2, 3, 5, 7, 11, 13, 17, 19.
Composite number: a number with more than two factors. First few: 4, 6, 8, 9, 10, 12…

Imp: The number 1 is neither prime nor composite (it has only one factor).

Sieve of Eratosthenes (Steps)

Cross out 1

➜

Circle 2, cross its multiples

➜

Circle 3, cross its multiples

➜

Repeat for 5, 7…

➜

Circled = Primes

All circled numbers are primes; crossed numbers (except 1) are composites. This method was created by the Greek mathematician Eratosthenes about 2200 years ago.

Figure it Out — Pages 113–115

Page 113 – Primes and composites from 21 to 30:

Ans: Primes → 23, 29 (2 numbers). Composites → 8 numbers.

1. Any other even prime besides 2?

Ans: No. 2 is the only even prime (every other even number is divisible by 2).

2. Smallest & largest gap between successive primes (up to 100):

Ans: Smallest = 1 (between 2 and 3). Largest = 8 (between 89 and 97).

3. Do all rows have equal primes? Which decades have most/least?

Ans: No. Fewest primes in 91–100; most in 1–10 and 11–20.

4. Which are prime: 23, 51, 37, 26?

Ans: 23 and 37.

5. Three prime pairs (below 20) whose sum is a multiple of 5:

Ans: (2, 3), (3, 7), (2, 13).

6. Prime pairs with same digits (like 13 & 31) up to 100:

Ans: 17 & 71, 37 & 73, 79 & 97.

7. Seven consecutive composite numbers below 100:

Ans: 90, 91, 92, 93, 94, 95, 96.

8. Twin primes (difference of 2) up to 100:

Ans: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43), (59,61), (71,73).

9. True or False:

a) No prime ends in 4 → True (ending 4 = even, only even prime is 2).
b) A product of primes can be prime → False (it becomes composite).
c) Primes have no factors → False (they have exactly two: 1 and itself).
d) All even numbers are composite → False (2 is even but prime).
e) After every prime except 2, the next number is composite → True (every other prime is odd, so next is even/composite).

10. Which is a product of exactly three distinct primes: 45, 60, 91, 105, 330?

Ans: 105 = 3 × 5 × 7.

11. Three-digit primes using 2, 4, 5 once each?

Ans: None (every arrangement is even or ends in 5, so divisible).

12. Primes p where 2p + 1 is also prime (five examples):

Ans: 5→11, 11→23, 23→47, 29→59, 41→83.

5.3 Co-prime Numbers

Co-prime numbers: two numbers that have no common factor other than 1. Jumpy (jump size > 1) cannot reach both — so such a pair is "safe".

Example: 15 and 39 share 3 → not co-prime. But 4 and 9 share only 1 → co-prime.

Co-prime Art (Thread on Pegs)

When the number of pegs and the thread-gap are co-prime, the thread touches every peg. When they share a factor, only some pegs are used.

Figure it Out — Pages 115–116

Page 115 – Which pairs are safe (co-prime)?

Ans: Safe → 4 & 15, 18 & 29. (Not safe → 15 & 39, 20 & 55.)

Page 116 – Which pairs are co-prime?

Ans: a) 18 & 35 ✓, b) 15 & 37 ✓, d) 17 & 69 ✓. (Not: c) 30 & 415 shares 5, e) 81 & 18 shares 9.)

First common multiple = product, or less?

Equal to product (co-prime): e.g. 3 & 5, 3 & 7, 4 & 9.
Less than product (not co-prime): e.g. 3 & 6, 3 & 12, 6 & 15.
Imp: When two numbers are co-prime, their first common multiple = their product.

5.4 Prime Factorisation

Writing a number as a factor pair (like 56 = 14 × 4) does not show all factors. To be sure whether two numbers share a factor, break each all the way down to primes.

Prime Factorisation: writing a number as a product of only prime numbers. Example: 56 = 2 × 2 × 2 × 7.

Factor Tree of 36

Imp: Every number greater than 1 has only one prime factorisation (order of primes may change, but the primes are the same). The number 1 has no prime factorisation.

Two Useful Ideas

1. Check co-prime: Find prime factors of both. No common prime factor → co-prime.
Example: 80 = 2×2×2×2×5, 63 = 3×3×7 → no common prime → co-prime.

2. Check divisibility: The first number is divisible by the second if the second's prime factorisation is fully inside the first's.
Example: 168 = 2×2×2×3×7 contains 12 = 2×2×3 → 168 is divisible by 12.

Figure it Out — Page 120

1. Prime factorisations:

Number	Prime Factorisation
64	2×2×2×2×2×2
104	2×2×2×13
105	3×5×7
243	3×3×3×3×3
320	2×2×2×2×2×2×5
141	3×47
1728	2×2×2×2×2×2×3×3×3
729	3×3×3×3×3×3
1024	2×2×2×2×2×2×2×2×2×2
1331	11×11×11
1000	2×2×2×5×5×5

2. Number with one 2, two 3s, one 11:

Ans: 2 × 3 × 3 × 11 = 198.

3. Three primes under 30 whose product is 1955:

Ans: 5 × 17 × 23 = 1955.

4. Prime factorisation without multiplying first:

a) 56 × 25 = 2×2×2×7×5×5
b) 108 × 75 = 2×2×3×3×3×3×5×5
c) 1000 × 81 = 2×2×2×3×3×3×3×5×5×5

5. Smallest number whose prime factorisation has:

a) three different primes → 2 × 3 × 5 = 30
b) four different primes → 2 × 3 × 5 × 7 = 210

Figure it Out — Page 122

1. Are these co-prime?

a) 30 & 45 → No (share 3, 5) b) 57 & 85 → Yes
c) 121 & 1331 → No (share 11) d) 343 & 216 → Yes

2. Is the first divisible by the second?

a) 225 & 27 → No b) 96 & 24 → Yes
c) 343 & 17 → No d) 999 & 99 → No

3. First = 2×3×7, second = 3×7×11. Co-prime? Does one divide the other?

Ans: Not co-prime (share 3 and 7). Neither divides the other.

4. Guna says any two primes are co-prime — right?

Ans: Yes. Two different primes have no common factor except 1.

5.5 Divisibility Tests

You can often tell if a large number is divisible without long division by looking at its last digit(s).

Divisor	Rule (look at…)	Divisible if last digit(s) is…
10	last 1 digit	0
5	last 1 digit	0 or 5
2	last 1 digit	0, 2, 4, 6, 8
4	last 2 digits	form a number divisible by 4
8	last 3 digits	form a number divisible by 8

Imp: A single last digit does not tell divisibility by 4 (12 vs 22). Check the last two digits for 4, and the last three digits for 8. Rules for 3, 6, 7, 9 come in later classes.

Figure it Out — Pages 124–126

Is 8536 divisible by 4?

Ans: Yes — last two digits 36 are divisible by 4. (All three statements about "last two digits" are Yes.)

Change last two digits of 8560 to make it a multiple of 8:

Ans: 8552 (552 ÷ 8 works).

Statements about last-three-digits for 8:

Ans: All three are Yes. Examples: 8576, 7648, 5024.

1. Leap years (multiple of 4, but a century year only if divisible by 400):

b) From 2024 to 2099 there are 19 leap years.

2. Largest & smallest 4-digit palindromes divisible by 4:

Ans: Largest = 8888, Smallest = 2112.
(A palindrome reads the same both ways; its last two digits must be divisible by 4 — 9999 and 1001 are not.)

3. Always / sometimes / never true:

a) Sum of two even numbers is a multiple of 4 → Sometimes (2+6=8 ✓, 2+4=6 ✗).
b) Sum of two odd numbers is a multiple of 4 → Sometimes (1+3=4 ✓, 1+5=6 ✗).

4. Remainders when divided by 10, 5, 2:

Number	÷10	÷5	÷2
78	8	3	0
99	9	4	1
173	3	3	1
572	2	2	0
980	0	0	0
1111	1	1	1
2345	5	0	1

5. Guna checks divisibility of 14560 by only two numbers to cover 2, 4, 5, 8, 10:

Ans: 5 and 8. (Divisible by 8 → also by 2 and 4; divisible by 5 and 2 → also by 10.)

6. Which are divisible by all of 2, 4, 5, 8, 10: 572, 2352, 5600, 6000, 77622160?

Ans: 5600, 6000, 77622160.

7. Two numbers with product 10000, neither ending in 0:

Ans: 16 × 625 = 10000.

5.6 Fun with Numbers

Special Numbers

For the box 9, 16, 25, 43, each number can be "special" for a different reason:

9 — only single-digit number; only multiple of 3.
16 — only even number; only multiple of 4.
25 — only multiple of 5.
43 — only prime; only number that is not a perfect square.

A Prime Puzzle

Rule: Fill the grid using only prime numbers so that the product of each row equals the number on its right, and the product of each column equals the number below it.

Worked grid:

5	5	3	75
2	3	7	42
17	2	3	102
170	30	63	✓

Check: row 1 → 5×5×3=75; column 1 → 5×2×17=170. All rows and columns match.

Imp Points for Exams

1 is neither prime nor composite.
2 is the only even prime number.
Every number > 1 has exactly one prime factorisation.
Co-prime ⇒ no common factor other than 1 ⇒ first common multiple = product.
Divisibility: last 1 digit for 2/5/10, last 2 for 4, last 3 for 8.