Chapter 5: Prime Time from the class 6 NCERT Ganita Prakash textbook.
5.1 Common Multiples and Common Factors
Page 108: Idli-Vada Game
1. At what number is ‘idli-vada’ said for the 10th time?
* Explanation: "Idli-vada" is said for common multiples of 3 and 5, which are multiples of their LCM, 15.
* Solution: The sequence is 15, 30, 45... This is an arithmetic progression. The 10th term is `15 + (10-1)*15 = 15 + 135 = 150`.
* Answer: `150`
2. If the game is played for the numbers 1 to 90, find out:
* a. How many times would the children say ‘idli’?
* Explanation: "Idli" is for multiples of 3.
* Solution: `90 ÷ 3 = 30`
* Answer: `30 times`
* b. How many times would the children say ‘vada’?
* Explanation: "Vada" is for multiples of 5.
* Solution: `90 ÷ 5 = 18`
* Answer: `18 times`
* c. How many times would the children say ‘idli-vada’?
* Explanation: "Idli-vada" is for multiples of 15.
* Solution: `90 ÷ 15 = 6`
* Answer: `6 times`
3. What if the game was played till 900? How would your answers change?
* Solution:
* Idli: `900 ÷ 3 = 300 times`
* Vada: `900 ÷ 5 = 180 times`
* Idli-Vada: `900 ÷ 15 = 60 times`
* Answer: The counts would be 300, 180, and 60 respectively.
4. Is this figure somehow related to the ‘idli-vada’ game?
* Explanation: The figure (Fig. 5.1) is a visual tool like a Venn diagram or number line.
* Answer: `Yes`. It is used to visually represent and identify multiples of one number, multiples of another, and their common multiples (the overlapping region), which are the "idli-vada" numbers.
Page 109: Idli-Vada with different pairs
*(This is an activity to draw a figure. The solution provides the data for the drawing.)*
Play the game with: a. 2 and 5, b. 3 and 7, c. 4 and 6. Draw a figure up to 60.
* Solution (Data for the figure):
* a. Pair (2,5):
* Multiples of 2 (Idli): 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60
* Multiples of 5 (Vada): 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60
* Common Multiples (Idli-Vada): 10, 20, 30, 40, 50, 60
* b. Pair (3,7):
* Multiples of 3 (Idli): 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60
* Multiples of 7 (Vada): 7, 14, 21, 28, 35, 42, 49, 56
* Common Multiples (Idli-Vada): 21, 42
* c. Pair (4,6):
* Multiples of 4 (Idli): 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60
* Multiples of 6 (Vada): 6, 12, 18, 24, 30, 36, 42, 48, 54, 60
* Common Multiples (Idli-Vada): 12, 24, 36, 48, 60
Pages 110-111: Figure it Out
1. Find all multiples of 40 that lie between 310 and 410.
* Solution: `320 (8x40), 360 (9x40), 400 (10x40)`
* Answer: `320, 360, 400`
2. Who am I?
* a. Less than 40, factor 7, sum of digits is 8.
* Solution: Multiples of 7 < 40: 7, 14, 21, 28, 35. Only `35` has digits 3+5=8.
* Answer: `35`
* b. Less than 100, factors 3 and 5, one digit is 1 more than the other.
* Solution: Multiples of 15 < 100: 15, 30, 45, 60, 75, 90. Only `45` has digits (4 and 5) differing by 1.
* Answer: `45`
3. Find a perfect number between 1 and 10.
* Explanation: A perfect number's factors (excluding itself) sum to the number.
* Solution: Number 6. Factors: 1, 2, 3, 6. Sum of factors (excluding 6) = 1+2+3 = 6.
* Answer: `6`
4. Find the common factors of:
* a. 20 and 28: Factors of 20 (1,2,4,5,10,20); of 28 (1,2,4,7,14,28). Common: `1, 2, 4`
* b. 35 and 50: Factors of 35 (1,5,7,35); of 50 (1,2,5,10,25,50). Common: `1, 5`
* c. 4, 8 and 12: Factors of 4 (1,2,4); of 8 (1,2,4,8); of 12 (1,2,3,4,6,12). Common: `1, 2, 4`
* d. 5, 15 and 25: Factors of 5 (1,5); of 15 (1,3,5,15); of 25 (1,5,25). Common: `1, 5`
5. Find any three numbers that are multiples of 25 but not multiples of 50.
* Explanation: These are the odd multiples of 25.
* Answer: `25, 75, 125` (Any three are acceptable).
6. Anshu plays the game with two numbers <10. First 'idli-vada' is after 50. What could the numbers be?
* Explanation: The first common multiple (LCM) must be >50.
* Solution: Check pairs: (7,8) LCM=56; (7,9) LCM=63; (8,9) LCM=72. All fit.
* Answer: Possible pairs are `7 and 8`, `7 and 9`, or `8 and 9`.
7. Treasures on 28 and 70. What jump sizes land on both?
* Explanation: Jump sizes are common factors.
* Solution: Factors of 28 (1,2,4,7,14,28); of 70 (1,2,5,7,10,14,35,70). Common: `1, 2, 7, 14`
* Answer: Jump sizes of `1, 2, 7, or 14`
8. Guna erased all numbers except the common multiples (24,48,72). Fill the figure.
* Explanation: The common multiples are of the LCM. 24 is the LCM. The two numbers could be factors of 24 whose LCM is 24, e.g., 6 and 8.
* Answer (One possible solution for numbers 6 & 8):
* Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72
* Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72
* Common Multiples (Intersection): 24, 48, 72
9. Smallest number that is a multiple of 1 to 10, except 7.
* Explanation: Find LCM of {1,2,3,4,5,6,8,9,10}.
* Solution: Prime powers needed: 2³ (from 8), 3² (from 9), 5 (from 5,10). LCM = 8 x 9 x 5 = `360`.
* Answer: `360`
10. Smallest number that is a multiple of all numbers from 1 to 10.
* Solution: LCM of 1-10. From Q9, add the factor 7. LCM = 360 x 7 = `2520`.
* Answer: `2520`
5.2 Prime Numbers
Pages 114-115: Figure it Out
1. Is there any other even prime?
* Answer: `No, 2 is the only even prime number.`
2. smallest and largest difference between successive primes till 100?
* Solution: Smallest difference is `2` (e.g., 3&5). Largest is `8` (between 89 and 97).
3. Equal primes in every decade? Most? Least?
* Answer: `No`, the number is not equal. The decade 10-19 has the most primes (4: 11,13,17,19). The decade 90-99 has the least (1: 97).
4. Which are prime: 23, 51, 37, 26?
* Solution: 23 (prime), 51 (divisible by 3), 37 (prime), 26 (divisible by 2).
* Answer: `23 and 37`
5. Three pairs of primes <20 whose sum is a multiple of 5.
* Solution: Primes <20: 2,3,5,7,11,13,17,19. Pairs: (2,3)=5, (3,7)=10, (3,17)=20, (13,17)=30.
* Answer: Three pairs: `(2,3)`, `(3,7)`, `(13,17)`
6. Pairs of primes up to 100 that are reversals (like 13,31).
* Answer: `(17, 71)`, `(37, 73)`, `(79, 97)`
7. Find seven consecutive composite numbers between 1 and 100.
* Answer: `90, 91, 92, 93, 94, 95, 96`
8. Find twin primes (difference 2) between 1 and 100.
* Answer: `(3,5)`, `(5,7)`, `(11,13)`, `(17,19)`, `(29,31)`, `(41,43)`, `(59,61)`, `(71,73)`
9. True or False?
* a. No prime ends with 4: `True` (All such numbers are even and >2).
* b. Product of primes can be prime: `False` (It will have at least two factors).
* c. Primes have no factors: `False` (They have two factors: 1 and itself).
* d. All even numbers are composite: `False` (2 is prime).
* e. After 2 & 3, the next number after a prime is always composite: `True` (Because primes >2 are odd, so the next number is even and greater than 2, hence composite).
10. Which number is a product of exactly three distinct primes: 45, 60, 91, 105, 330?
* Solution:
* 45 = 3²×5 (2 distinct)
* 60 = 2²×3×5 (3 distinct) tick mark
* 91 = 7×13 (2 distinct)
* 105 = 3×5×7 (3 distinct) tick mark
* 330 = 2×3×5×11 (4 distinct)
* Answer: `60 and 105`
11. How many 3-digit primes can be made from 2,4,5 used once?
* Explanation: All numbers will end in 2,4, or 5, making them divisible by 2 or 5.
* Answer: `None (0)`
12. Find primes `p` such that `2p + 1` is also prime (like 3→7).
* Answer: Five examples: `(2→5)`, `(5→11)`, `(11→23)`, `(23→47)`, `(29→59)`
5.3 Co-prime Numbers & 5.4 Prime Factorisation
*(Note: Pages 116-117 are primarily explanatory. The main problems for this section are on Pages 122.)*
Page 122: Figure it Out
1. Are these pairs co-prime?
* a. 30 and 45: Prime factors: 30=2×3×5; 45=3²×5. Common factors 3 and 5. Not co-prime.
* b. 57 and 85: 57=3×19; 85=5×17. No common prime factors. Co-prime.
* c. 121 and 1331: 121=11²; 1331=11³. Common factor 11. Not co-prime.
* d. 343 and 216: 343=7³; 216=2³×3³. No common prime factors. Co-prime.
2. Is the first number divisible by the second?
* a. 225 and 27: 225=3²×5²; 27=3³. 27's prime factorisation (3³) is *not* included in 225's (3²×5²), as 225 only has two 3's. No.
* b. 96 and 24: 96=2⁵×3; 24=2³×3. 24's factors are included in 96's. Yes (96 ÷ 24 = 4).
* c. 343 and 17: 343=7³; 17 is prime. 17 is not a factor of 343. No.
* d. 999 and 99: 999=3³×37; 99=3²×11. The factor 11 from 99 is not present in 999. No.
3. First number: 2×3×7. Second: 3×7×11. Co-prime? Divisible?
* Co-prime? They share common prime factors 3 and 7. Not co-prime.
* Divisible? The first number (42) does not contain the prime factor 11 from the second number (231). So, 231 does not divide 42. Conversely, 42's factors (2,3,7) are all in 231, but 231 has an *extra* 11. So, 42 divides 231? 231 ÷ 42 = 5.5, which is not an integer. Neither number divides the other.
4. Guna says, “Any two prime numbers are co-prime?”. Is he right?
* Answer: `Yes`. The only common factor two different prime numbers can have is 1. (If they are the same prime, the common factors are 1 and the prime itself, so they are not co-prime).
5.5 Divisibility Tests
Pages 125-126: Figure it Out
4. Find remainders when divided by 10, 5, 2.
* Explanation: Remainder when divided by:
* 10 is the last digit.
* 5 is the last digit (if last digit <5, it is the remainder; if >=5, subtract 5).
* 2 is 0 if even, 1 if odd.
* Solution:
* 78: ÷10=8; ÷5=3 (5 fits in 75, remainder 3); ÷2=0.
* 99: ÷10=9; ÷5=4 (95+4); ÷2=1.
* 173: ÷10=3; ÷5=3 (170+3); ÷2=1.
* 572: ÷10=2; ÷5=2; ÷2=0.
* 980: ÷10=0; ÷5=0; ÷2=0.
* 1111: ÷10=1; ÷5=1; ÷2=1.
* 2345: ÷10=5; ÷5=0; ÷2=1.
5. Teacher asked if 14560 is divisible by 2,4,5,8,10. Guna checked only two numbers. Which two?
* Explanation: If a number is divisible by 8 and 5 (or 10), it is automatically divisible by 2 and 4 (since 8 is a multiple of 2 and 4). Divisibility by 5 and 8 implies divisibility by their LCM, 40, which doesn't guarantee divisibility by 10. However, since 14560 ends with a 0, it is divisible by 10. The most efficient check is for 8 and 5 (or 10).
* Answer: He checked divisibility by `8` and `5` (or `8` and `10`).
6. Which numbers are divisible by 2,4,5,8,10?
* Explanation: A number divisible by all these must be divisible by their LCM, which is 40. But for divisibility by 2,4,5,8,10, the number must end with a 0 and the last 3 digits must form a number divisible by 8.
* Solution:
* 572: Last digit 2 (div by 2), not 0 → fails 5,10.
* 2352: Last digit 2 → fails 5,10.
* 5600: Ends with 00. Last 3 digits=600. 600÷8=75. Yes.
* 6000: Ends with 000. 0÷8=0. Yes.
* 77622160: Ends with 160. 160÷8=20. Yes.
* Answer: `5600, 6000, 77622160`
7. Write two numbers whose product is 10000. The two numbers should not have 0 as the units digit.
* Explanation: We need factors of 10000 that are not multiples of 10.
* Solution: 10000 = 2⁴ × 5⁴. We can group these primes so that neither group ends with 0 (i.e., neither has both a 2 and a 5).
* Option 1: (2⁴) and (5⁴) -> 16 and 625. (16 * 625 = 10000)
* Option 2: (2³×5) and (2×5³) -> 40 and 250 -> *Invalid, both end with 0.*
* Option 3: (2²×5²) and (2²×5²) -> 100 and 100 -> *Invalid.*
* Option 4: (2⁴×5) and (5³) -> 80 and 125 -> *80 ends with 0.*
* Option 5: (5⁴) and (2⁴) -> same as option 1.
* Answer: `16 and 625`
Of course. Here is the solution and explanation for the Prime Puzzle on Pages 127 and 128.
The Prime Puzzle
Rules: Fill the grid with prime numbers only so that the product of the numbers in each row is the number to the right of the row, and the product of the numbers in each column is the number below the column.
The puzzle grid is:
[ ] [ ] [ ] | 105
[ ] [ ] [ ] | 20
[ ] [ ] [ ] | 30
-----------------------
28 125 18
Solution:
Let's denote the grid as follows:
[ a ] [ b ] [ c ] | 105
[ d ] [ e ] [ f ] | 20
[ g ] [ h ] [ i ] | 30
-----------------------
28 125 18
Step 1: Find the Prime Factorizations of the Results
* Row 1 Product = 105: `105 = 3 × 5 × 7`
* Row 2 Product = 20: `20 = 2 × 2 × 5`
* Row 3 Product = 30: `30 = 2 × 3 × 5`
* Column 1 Product = 28: `28 = 2 × 2 × 7`
* Column 2 Product = 125: `125 = 5 × 5 × 5`
* Column 3 Product = 18: `18 = 2 × 3 × 3`
Step 2: Analyze Column 2 (Product = 125)
* The prime factorization of 125 is `5 × 5 × 5`.
* This means the three cells in Column 2 (b, e, h) must all be the prime number 5.
We can now fill these in:
[ a ] [ 5 ] [ c ] | 105
[ d ] [ 5 ] [ f ] | 20
[ g ] [ 5 ] [ i ] | 30
-----------------------
28 125 18
Step 3: Analyze Row 2 (Product = 20)
* We already have one cell: `e = 5`.
* The product of the other two cells (`d` and `f`) must be `20 ÷ 5 = 4`.
* The only way to get a product of 4 using two prime numbers is `2 × 2`.
* Therefore, `d = 2` and `f = 2`.
Let's fill these in:
[ a ] [ 5 ] [ c ] | 105
[ 2 ] [ 5 ] [ 2 ] | 20
[ g ] [ 5 ] [ i ] | 30
-----------------------
28 125 18
Step 4: Analyze Column 1 (Product = 28)
* We have two cells: `d = 2` and `g` is unknown.
* The product of the column is `2 × g × a = 28`.
* We already know from the factorization that 28 = `2 × 2 × 7`.
* We already have one `2` from cell `d`. Therefore, the product of `a` and `g` must be `2 × 7`.
* The only way to get this product with two primes is `2 × 7`.
* This means one of `a` or `g` is 2, and the other is 7.
* Let's keep this for now and use another column to decide.
Step 5: Analyze Column 3 (Product = 18)
* We have two cells: `f = 2` and `i` is unknown.
* The product of the column is `c × 2 × i = 18`.
* We know 18 = `2 × 3 × 3`.
* We already have one `2` from cell `f`. Therefore, the product of `c` and `i` must be `3 × 3`.
* The only way to get this product with two primes is `3 × 3`.
* Therefore, both `c = 3` and `i = 3`.
Let's fill these in:
[ a ] [ 5 ] [ 3 ] | 105
[ 2 ] [ 5 ] [ 2 ] | 20
[ g ] [ 5 ] [ 3 ] | 30
-----------------------
28 125 18
Step 6: Finalize Row 1 and Row 3
* Row 1 Product (a × 5 × 3 = 105): `a × 15 = 105`, so `a = 105 ÷ 15 = 7`.
* Row 3 Product (g × 5 × 3 = 30): `g × 15 = 30`, so `g = 30 ÷ 15 = 2`.
Now we can fill in the last two cells, `a` and `g`. From Step 4, we see this also satisfies the condition for Column 1 (7 × 2 × 2 = 28).
Step 7: Complete the Grid and Verify
The final filled grid is:
[ 7 ] [ 5 ] [ 3 ] | 105 (7 × 5 × 3 = 105 tick)
[ 2 ] [ 5 ] [ 2 ] | 20 (2 × 5 × 2 = 20 tick)
[ 2 ] [ 5 ] [ 3 ] | 30 (2 × 5 × 3 = 30 tick)
-----------------------
28 125 18
(7×2×2=28 tick) (5×5×5=125 tick) (3×2×3=18 tick)
Final Answer:
The solved prime puzzle is:
| 7 | 5 | 3 | 105
| 2 | 5 | 2 | 20
| 2 | 5 | 3 | 30
----------------------
28 125 18
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