Class 6 Maths Chapter 5 NCERT Solutions Hinglish mein: Prime Time

Ganita Prakash (Reprint 2026-27) ke Chapter 5, Prime Time, ke har “Figure it Out”, Math Talk aur Try This sawaal ke jawab, factor trees, sieve aur prime puzzles bane hue. Page numbers book se match karte hain, aur jawab chapter ke saath chhape NCERT ke apne solutions se milaaye gaye hain; jahan hum alag hain, wahan wajah bataayi hai. Apni zindagi waale sawaalon, jaise tumhare janm ka saal, ke sample jawab hain. Short notes: Chapter 5 notes.

Idli-vada game (page 107)

Jo numbers 3 aur 5 dono ke multiple hain, wo 15, 30, 45, 60, … hain. Inhe 3 aur 5 ke common multiples kehte hain.

Figure it Out: idli-vada (page 108)

1. ‘Idli-vada’ 3 aur 5 ke common multiples par bola jaata hai, jo 15 ke multiples hain. 10vi baar 15 × 10 = 150 par.

2. 1 se 90 tak: a. ‘idli’ (3 ke multiples, ‘idli-vada’ milaakar): 90 ÷ 3 = 30 baar. b. ‘vada’ (5 ke multiples): 90 ÷ 5 = 18 baar. c. ‘idli-vada’ (15 ke multiples): 90 ÷ 15 = 6 baar.

3. 900 tak har jawab 10 guna ho jaata hai: 300, 180 aur 60.

4. Haan. Ye figure 30 tak khela gaya game hai: baayein gole mein ‘idli’ numbers (3 ke multiples), daayein gole mein ‘vada’ numbers (5 ke multiples), aur beech mein ‘idli-vada’ numbers, 15 aur 30. (Book ke figure mein 6 chhoot gaya hai, jo 3 ke multiples mein aana chahiye.) 60 tak:

Doosre jode, 60 tak (‘idli-vada’ numbers):

• a. 2 aur 5: 10, 20, 30, 40, 50, 60 (10 ke multiples).
• b. 3 aur 7: 21 aur 42.
• c. 4 aur 6: 12, 24, 36, 48 aur 60 (12 ke multiples).

Kisi ne sirf ‘vada’ nahi bola (page 109): bade number ka har multiple chhote number ka bhi multiple tha, yaani bada number chhote ka multiple hai. Ek number 4 ho to doosra 2 ho sakta hai (4 ka har multiple 2 ka multiple hai) ya 8 (8 ka har multiple 4 ka multiple hai). 3, 5 ya 10 ke saath kabhi na kabhi sirf ‘vada’ aata, jaise 3 ke saath 4, ya 4 ke saath 5. (NCERT ki key sirf 8 deti hai, par 2 bhi niyam par fit hai, kyunki ‘vada’ bade number ke saath hai aur 4 ka har multiple 2 ka multiple hai.)

Jump Jackpot (pages 109–110)

15 aur 30 dono tak pahunchne waale jump size: 15 aur 30 ke common factors, yaani 1, 3, 5 aur 15.

31 se 70 ki table: 1. Shaded numbers sab 3 ke multiples hain. 2. Gole waale numbers sab 4 ke multiples hain. 3. 36, 48 aur 60 shaded bhi hain aur gole mein bhi; ye 3 aur 4 ke common multiples hain.

Figure it Out: multiples aur factors (pages 110–111)

1. 310 aur 410 ke beech 40 ke multiples: 320, 360 aur 400.

2. a. 40 se chhote 7 ke multiples 7, 14, 21, 28 aur 35 hain; sirf 35 ke ankon ka jod 8 hai: 35. b. 100 se chhote numbers jinke factors 3 aur 5 hon, 15 ke multiples hain: 15, 30, 45, 60, 75, 90; sirf 45 mein ek ank doosre se 1 zyada hai: 45.

3. 6 perfect number hai: iske factors 1, 2, 3 aur 6 ka jod 12 hai, jo 6 ka dugna hai.

4. a. 20 aur 28: 1, 2, 4. b. 35 aur 50: 1, 5. c. 4, 8 aur 12: 1, 2, 4. d. 5, 15 aur 25: 1, 5.

5. 25 ke odd multiples: 25, 75 aur 125 (aur 175, 225, …).

6. Dono numbers ka pehla common multiple 50 se zyada hona chahiye. 10 se chhote numbers mein aisa sirf 7 aur 8 (pehla ‘idli-vada’ 56 par), 7 aur 9 (63) aur 8 aur 9 (72) ke saath hota hai.

7. 28 aur 70 ke common factors: jump size 1, 2, 7 aur 14.

8. Common multiples 24, 48 aur 72, 24 ke multiples hain, isliye dono numbers ka pehla common multiple 24 hai. Ek jawab 6 aur 8 hai. 72 tak sirf 6 ke multiples 6, 12, 18, 30, 36, 42, 54, 60 aur 66; sirf 8 ke multiples 8, 16, 32, 40, 56 aur 64; aur common multiples 24, 48 aur 72. Doosre sambhav jode 3 aur 8, aur 8 aur 12 hain. (NCERT ki key 3 aur 4 sujhaati hai, par tab 12, 36 aur 60 bhi common multiples hote, jabki diagram mein sirf 24, 48 aur 72 hain.)

9. Number mein 7 ko chhodkar 1 se 10 tak ke primes hone chahiye: teen 2 (8 ke liye), do 3 (9 ke liye) aur ek 5, yaani 2 × 2 × 2 × 3 × 3 × 5 = 360.

10. 7 bhi jodo: 360 × 7 = 2520.

Prime numbers (pages 112–113)

Anjeer lagaana: 12 anjeer 1 × 12, 2 × 6, 3 × 4, 4 × 3, 6 × 2 aur 12 × 1 mein lag sakte hain; 7 anjeer sirf 1 × 7 ya 7 × 1 mein. Kataaron aur columns ki ginti hamesha anjeer ki ginti ke factors hoti hai.

21 se 30 tak: 2 primes (23 aur 29) aur 8 composite numbers (21, 22, 24, 25, 26, 27, 28 aur 30).

Sieve se 1 se 100 tak ke primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 aur 97, yaani 25 primes.

Sieve kaise kaam karta hai: har kata number kisi chhote prime ka multiple hai, isliye uska 1 aur khud ke alawa bhi factor hai aur wo composite hai. Jo number kabhi nahi kat-ta, wo kisi chhote prime ka multiple nahi, isliye uske factors sirf 1 aur khud hain: wo prime hai.

Figure it Out: primes (pages 114–115)

1. Nahi. 2 se bade har even number ka factor 2 hai, isliye uske do se zyada factors hain.

2. Sabse chhota farq: 1 (2 aur 3). 100 tak sabse bada farq: 8 (89 aur 97).

3. Nahi. Har row mein primes: 1–10: 4; 11–20: 4; 21–30: 2; 31–40: 2; 41–50: 3; 51–60: 2; 61–70: 2; 71–80: 3; 81–90: 2; 91–100: 1. Sabse kam 91–100 mein (sirf 97); sabse zyada 1–10 aur 11–20 mein (4-4).

4. 23 aur 37 prime hain; 51 = 3 × 17 aur 26 = 2 × 13 nahi.

5. Jaise 2 + 3 = 5, 3 + 7 = 10 aur 11 + 19 = 30 (aur 2 + 13, 3 + 17, 7 + 13, 13 + 17).

6. 13 aur 31, 17 aur 71, 37 aur 73, 79 aur 97.

7. 90, 91, 92, 93, 94, 95, 96, primes 89 aur 97 ke beech.

8. 3 aur 5, aur 17 aur 19 ke alawa: 5 aur 7, 11 aur 13, 29 aur 31, 41 aur 43, 59 aur 61, 71 aur 73.

9. a. Sahi: 4 par khatm hone waala number even aur 2 se bada hota hai. b. Galat: do ya zyada primes ke guna ke factors wo primes bhi hote hain, isliye uske do se zyada factors hote hain. c. Galat: prime ke theek do factors hote hain, 1 aur khud. d. Galat: 2 even hai aur prime bhi. e. Sahi: 2 ke baad har prime odd hai, isliye agla number even aur 2 se bada, yaani composite hai.

10. 105 = 3 × 5 × 7. (45 = 3 × 3 × 5 aur 91 = 7 × 13 mein do primes hain; 60 = 2 × 2 × 3 × 5 mein 2 dohraata hai; 330 = 2 × 3 × 5 × 11 mein chaar hain.)

11. Ek bhi nahi. 245, 254, 425, 452, 524 aur 542 sab 2, 4 ya 5 par khatm hote hain, isliye har ek 2 ya 5 se divisible hai.

12. Haan: 2 → 5, 5 → 11, 11 → 23, 23 → 47, 29 → 59 (aur 41 → 83 aur 53 → 107).

Co-prime numbers (pages 115–117)

Surakshit jode: a. 15 aur 39: surakshit nahi (3 ki chhalaang). b. 4 aur 15: surakshit. c. 18 aur 29: surakshit. d. 20 aur 55: surakshit nahi (5 ki chhalaang).

Co-prime jode: a. 18 aur 35: co-prime. b. 15 aur 37: co-prime. c. 30 aur 415: nahi (dono 5 se divisible). d. 17 aur 69: co-prime (69 = 3 × 23). e. 81 aur 18: nahi (dono 9 se divisible).

Anshu ka observation: 1. Jab numbers co-prime hon, to pehla common multiple unka guna hota hai: 3 aur 5 pehli baar 15 par milte hain, aur 4 aur 9, 36 par. 2. Jab koi factor common ho, to ye guna se chhota hota hai: 4 aur 6 pehli baar 12 par milte hain (guna 24), aur 6 aur 8, 24 par (guna 48).

Co-prime thread art: dhaaga har kheel tak tabhi pahunchta hai jab kheelon ki ginti aur thread-gap co-prime hon; warna wo jaldi band ho jaata hai. a. 15 kheelein, gap 10 (common factor 5): sirf 3 kheelein, ek tribhuj. b. 10 kheelein, gap 7: co-prime, saari 10. c. 14 kheelein, gap 6 (common factor 2): sirf 7 kheelein. d. 8 kheelein, gap 3: co-prime, saari 8.

Prime factorisation (pages 117–120)

Anshu ki galti: ek factor joda saare factors nahi dikhaata; 56 = 7 × 8 aur 63 = 9 × 7 se common factor 7 dikh jaata hai.

72 = 12 × 6: 72 = 2 × 2 × 3 × 2 × 3 = 2 × 2 × 2 × 3 × 3. 2 teen baar aata hai (do baar 12 mein, ek baar 6 mein) aur 3 do baar (har ek mein ek baar), theek 12 aur 6 ko milaakar jitna.

30 = 2 × 3 × 5 kisi bhi kram mein kyun: 30 cubes 2 oonche, 3 chaude aur 5 lambe dabbe ki tarah hain. Inhe 15-15 ki 2 parton, 10-10 ki 3 parton ya 6-6 ki 5 parton mein gino, hamesha 30 aata hai, isliye guna ke kram se farq nahi padta.

Figure it Out (page 120)

1.

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	das 2: 2 × 2 × … × 2
1331	11 × 11 × 11
1000	2 × 2 × 2 × 5 × 5 × 5

2. 2 × 3 × 3 × 11 = 198.

3. 1955 = 5 × 391 = 5 × 17 × 23.

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

5. a. 2 × 3 × 5 = 30. b. 2 × 3 × 5 × 7 = 210.

Composite common factor? Agar 80 aur 63 ka koi composite common factor hota, to uske prime factors dono ki prime factorisation mein aate. Kyunki 80 = 2 × 2 × 2 × 2 × 5 aur 63 = 3 × 3 × 7 mein koi prime common nahi, unka 1 ke alawa koi common factor nahi: wo co-prime hain.

Figure it Out: co-primes aur divisibility (page 122)

1. a. 30 = 2 × 3 × 5 aur 45 = 3 × 3 × 5: co-prime nahi (3 aur 5 common). b. 57 = 3 × 19 aur 85 = 5 × 17: co-prime. c. 121 = 11 × 11 aur 1331 = 11 × 11 × 11: co-prime nahi. d. 343 = 7 × 7 × 7 aur 216 = 2 × 2 × 2 × 3 × 3 × 3: co-prime.

2. a. 225 = 3 × 3 × 5 × 5 aur 27 = 3 × 3 × 3: Nahi, 225 mein sirf do 3 hain. b. 96 = 2 × 2 × 2 × 2 × 2 × 3 aur 24 = 2 × 2 × 2 × 3: Haan, 96 = 24 × 4. c. 343 = 7 × 7 × 7 aur 17 prime hai: Nahi. d. 999 = 3 × 3 × 3 × 37 aur 99 = 3 × 3 × 11: Nahi, 11, 999 ka factor nahi. (Key mein 999 ke liye chaar 3 likhe hain, par 3 × 3 × 3 × 37 = 999.)

3. Numbers 42 aur 231 hain. Inme 3 aur 7 common hain, isliye ye co-prime nahi. Koi doosre ko divide nahi karta: 42 mein ek 2 hai jo 231 mein nahi, aur 231 mein 11 hai jo 42 mein nahi.

4. Haan, do alag primes ke liye: har ek ke factors sirf 1 aur khud hain, isliye common factor sirf 1 hai. (Ek prime aur wahi prime, jaise 5 aur 5, ka common factor 5 hai.)

Divisibility tests (pages 122–125)

2 se 10 tak 8560 ke factors: 2, 4, 5, 8 aur 10. Ye 3, 6, 7 ya 9 se divisible nahi.

10 se: 125, 10 ka multiple nahi, kyunki 10 ke multiples 0 par khatm hote hain; 8560 hai. Kathan sahi hai: 10 se divisible numbers 0 par khatm hote hain.

5 se: 5, 10, 15, 20, … ke aakhri ank baari-baari 5 aur 0 hain. 399 se chhota sabse bada 5 se divisible number 395 hai; 8560, 5 se divisible hai. Kathan sahi hai.

2 se: 2 ke multiples 0, 2, 4, 6 ya 8 par khatm hote hain, isliye 682 aur 8560, 2 se divisible hain. 399 aur 411 ke beech 2 ke multiples: 400, 402, 404, 406, 408, 410. Kathan sahi hai.

4 se: 330 se 340 tak 4 ke multiples: 332, 336; 1730 se 1740: 1732, 1736; 2030 se 2040: 2032, 2036. Aakhri do ank dohraate hain, isliye sirf wahi maayne rakhte hain. 8536, 4 se divisible hai (36 hai). Teeno kathan sahi hain, kyunki 100, 4 ka multiple hai.

8 se: 120 se 140: 128, 136; 1120 se 1140: 1128, 1136; 3120 se 3140: 3128, 3136. Aakhri do ank kaafi nahi (200, 8 ka multiple hai par 100 nahi), par aakhri teen kaafi hain, kyunki 1000, 8 ka multiple hai. 8560 ke aakhri do ank badalkar 8 ke doosre multiples bante hain, jaise 8568 ya 8576; 8560 khud bhi ek hai (560 = 8 × 70). Teeno kathan sahi hain.

Figure it Out: divisibility (pages 125–126)

1. a. Sample: 2014 mein paida hue bachche ke liye ab tak ke leap years 2016, 2020 aur 2024 hain. b. 2024 se 2099 tak: 2024, 2028, …, 2096, yaani 19 leap years.

2. Sabse bada: 8888; sabse chhota: 2112. (Pehla ank aakhri ke barabar hai, jo even hona chahiye, aur aakhri do ankon se 4 ka multiple banna chahiye. NCERT ki key 9999 aur 1001 deti hai, par dono odd hain, isliye koi bhi 4 se divisible nahi.)

3. a. Kabhi-kabhi sahi: 2 + 2 = 4 hai, 2 + 4 = 6 nahi. b. Kabhi-kabhi sahi: 1 + 3 = 4 hai, 1 + 5 = 6 nahi.

4.

Number	10, 5 aur 2 se bhaag par shesh
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. 8 aur 5 (ya 8 aur 10). 8 se divisible matlab 4 aur 2 se bhi, aur 2 aur 5 se divisible matlab 10 se bhi.

6. 5600, 6000 aur 77622160. (572 aur 2352, 0 ya 5 par khatm nahi hote.)

7. 16 × 625 = 10000.

Numbers ke saath mazaa (pages 126–128)

Khaas numbers (sample kaaran):

• 5, 7, 12, 35: 5 sabse chhota hai aur akela jo 4 se bhaag dene par 1 bachaata hai; 7 akela jo 3 se bhaag dene par 1 bachaata hai; 12 akela even number hai; 35 akela jo box ke do numbers ka guna hai (5 × 7).
• 3, 8, 11, 24: 3 akela odd ek ank waala number hai; 8 akela cube hai (2 × 2 × 2); 11 akela do ank ka prime hai aur 3 + 8 ke barabar; 24 akela jo 3 aur 8 dono ka multiple hai (3 × 8).
• 27, 3, 123, 31: 3 akela ek ank ka number hai; 27 akela cube hai; 123 akela teen ank ka number hai; 31 akela jo 3 ka multiple nahi.
• 17, 27, 44, 65: 17 akela prime hai; 27 akela 3 ka multiple hai; 44 akela even number hai; 65 akela 5 ka multiple hai.

Prime puzzles: har puzzle ka theek ek jawab hai.

Aur practice ke liye padho prime aur composite numbers, co-prime numbers, prime factorisation kaise karein aur divisibility ke rules.