Class 7 Maths Part 2 Chapter 3 NCERT Solutions Hinglish mein: Finding Common Ground

Ganita Prakash Grade 7 (2026-27) ke Part 2 Chapter 3, Finding Common Ground, ke har sawaal ka jawab: text ke beech ke sawaal, har Figure it Out, Try This aur Math Talk sawaal, aur Mystery Colours puzzle. Page numbers book ke hisaab se hain (pages 47–66). Short notes: Part 2 Chapter 3 ke notes.

Sabse bada (pages 47–48)

Sameeksha ko sambhav tiles mein sabse badi kyun leni chahiye? Badi tile zyaada farsh dhakti hai, isliye kam tiles lagti hain. 1 ft, 2 ft aur 4 ft ki tiles mein 192, 48 aur 12 tiles lagengi.

Use 4 ft ki kitni tiles khareedni chahiye? Chaudai mein 12 ft ÷ 4 ft = 3 tiles aur lambai mein 16 ft ÷ 4 ft = 4 tiles, isliye 3 × 4 = 12 tiles.

Agar side foot ka fraction ho sakti to kya jawab badalta? Nahi. Side 12 ft aur 16 ft dono mein poori baar fit honi chahiye. 4/3 ft ya 2/3 ft jaisi sides bhi fit hoti hain, par wo 4 ft se chhoti hain, isliye 4 ft hi sabse badi tile hai aur 12 tiles hi sabse kam.

84 kg aur 108 kg chawal ke liye Lekhana kis wazan ka bag chune? Wazan 84 aur 108 ka common factor hona chahiye (1, 2, 3, 4, 6 ya 12 kg). Sabse bhaari bag se sabse kam bags lagte hain, isliye 12 kg ke bags: 84 ÷ 12 = 7 aur 108 ÷ 12 = 9, kul 16 bags.

Jump Jackpot: 0 se dono khazaanon pe utarne waali sabse lambi jump. (a) 14 aur 30: 2. (b) 7 aur 11: 1. (c) 30 aur 50: 10. (d) 28 aur 42: 14.

Kya sabse lambi jump HCF hai? Kyun? Haan. 0 se shuru karke koi jump size sirf apne multiples pe utarta hai, isliye wo dono khazaanon tak tabhi pahunchega jab wo dono numbers ka factor, yaani common factor, ho. Sabse lambi aisi jump highest common factor hai.

Prime factorisation (pages 49–51)

Kya 90 ko 3 × 30 ya 2 × 45 mein todne se wahi primes milte hain? Haan. 90 ko kaise bhi todo, aakhir mein 2 × 3 × 3 × 5 hi aata hai; sirf kram badal sakta hai.

Dono figures se 105 aur 30 ka prime factorisation likho. 105 = 3 × 5 × 7 aur 30 = 2 × 3 × 5.

Division method se 1200 ka prime factorisation nikaalo. 1200 ÷ 2 = 600, ÷ 2 = 300, ÷ 2 = 150, ÷ 2 = 75, ÷ 3 = 25, ÷ 5 = 5, ÷ 5 = 1. Isliye 1200 = 2 × 2 × 2 × 2 × 3 × 5 × 5.

Kaunsa tareeka aasaan hai? Division method. Har kadam mein ek chhote prime se bhaag dena hai, kuch andaaza nahi lagaana padta. 1200 ko 40 × 30, phir 5 × 8 × 5 × 6 waghairah mein todne mein ek saath kai composite numbers todne padte hain, aur ek chhootna aasaan hai.

840 = 2 × 2 × 2 × 3 × 5 × 7. Kya 2 × 7 = 14 factor hai? Haan, 2 × 7 factorisation ka hissa hai; baaki 2 × 2 × 3 × 5 = 60 se 14 × 60 = 840. Kya 2 × 2 × 2 = 8 factor hai? Haan, 840 mein teen 2 hain; 8 × 105 = 840. Kya 3 × 3 × 3 = 27 factor hai? Nahi, 840 mein sirf ek 3 hai, isliye 27 uska hissa nahi ho sakta.

Jaancho ki 225 ka har factor list 1, 3, 5, 9, 15, 25, 45, 75, 225 mein hai. 225 ke factor jode 1 × 225, 3 × 75, 5 × 45, 9 × 25 aur 15 × 15 hain, jo theek yahi nau numbers dete hain.

Figure it Out (page 51)

Saare factors likho. (a) 90 = 2 × 3 × 3 × 5: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 (12 factors). (b) 105 = 3 × 5 × 7: 1, 3, 5, 7, 15, 21, 35, 105 (8 factors). (c) 132 = 2 × 2 × 3 × 11: 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132 (12 factors). (d) 360 = 2 × 2 × 2 × 3 × 3 × 5: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360 (24 factors). (e) 840 = 2 × 2 × 2 × 3 × 5 × 7: 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840 (32 factors).

(Extra) Ginti ka ek pattern hai: har prime jitni baar aata hai usme 1 jodo aur guna karo. 360 ke liye (3 + 1) × (2 + 1) × (1 + 1) = 24.

Anshu kehta hai ki bade number ka prime factorisation hamesha lamba hota hai. Kya wo sahi hai? Nahi. Uska daava ek conjecture hai, aur ek counterexample use galat saabit karta hai: 96 = 2 × 2 × 2 × 2 × 2 × 3 mein chhe primes hain, par bade 121 = 11 × 11 mein sirf do. Isi tarah 64 mein chhe primes hain jabki bade 97, jo prime hai, mein ek.

Prime factors se HCF (pages 52–53)

Example 1: 45 = 3 × 3 × 5 aur 75 = 3 × 5 × 5 ke saajha hisse likho. 3, 5 aur 3 × 5 = 15. 1 ke saath ye common factors hain, aur HCF 15 hai.

Common factors aur HCF nikaalo. (a) 50 = 2 × 5 × 5 aur 60 = 2 × 2 × 3 × 5: common factors 1, 2, 5, 10; HCF = 10. (b) 140 = 2 × 2 × 5 × 7 aur 275 = 5 × 5 × 11: common factors 1, 5; HCF = 5. (c) 77 = 7 × 11 aur 725 = 5 × 5 × 29: koi prime saajha nahi, isliye ek hi common factor 1; HCF = 1. (d) 370 = 2 × 5 × 37 aur 592 = 2 × 2 × 2 × 2 × 37: common factors 1, 2, 37, 74; HCF = 74. (e) 81 = 3 × 3 × 3 × 3 aur 243 = 3 × 3 × 3 × 3 × 3: common factors 1, 3, 9, 27, 81; HCF = 81 (81, 243 ka factor hai).

Figure it Out (page 54)

1. HCF nikaalo. Saajha primes rakho, har ek sabse kam baar. (a) 24 = 2 × 2 × 2 × 3, 180 = 2 × 2 × 3 × 3 × 5: HCF = 2 × 2 × 3 = 12. (b) 42 = 2 × 3 × 7, 75 = 3 × 5 × 5, 24 = 2 × 2 × 2 × 3: teeno mein sirf 3 hai, isliye HCF = 3. (c) 240 = 2 × 2 × 2 × 2 × 3 × 5, 378 = 2 × 3 × 3 × 3 × 7: HCF = 2 × 3 = 6. (d) 400 = 2 × 2 × 2 × 2 × 5 × 5, 2500 = 2 × 2 × 5 × 5 × 5 × 5: HCF = 2 × 2 × 5 × 5 = 100. (e) 300 = 2 × 2 × 3 × 5 × 5, 800 = 2 × 2 × 2 × 2 × 2 × 5 × 5: HCF = 2 × 2 × 5 × 5 = 100.

2. 72 = 6 × 12 aur 144 = 8 × 18 dekhkar kya keh sakte hain ki 72 aur 144 mein 1 ke alaava koi factor saajha nahi? Nahi. Ye composite numbers mein tode gaye hain, primes mein nahi, aur composite tukde saajha primes chhupa lete hain. Primes mein 72 = 2 × 2 × 2 × 3 × 3 aur 144 = 2 × 2 × 2 × 2 × 3 × 3, isliye saajha hissa 2 × 2 × 2 × 3 × 3 = 72 hai. Asal mein 144 = 2 × 72, aur HCF 72 hai.

Sabse chhota, par aakhri nahi (pages 55–56)

Anshu (6 cm pattiyan) aur Guna (8 cm pattiyan) dono jo sabse chhota toran bana sakte hain, wo kitna lamba hai? Lambai 6 aur 8 ka common multiple honi chahiye; common multiples 24, 48, 72 aur aage hain, isliye sabse chhota 24 cm.

Kya sabse bada common multiple hota hai? Nahi. Agar koi number common multiple hai, to uska dugna, tigna aur aage bhi common multiples hain, isliye hamesha ek aur bada milta hai.

Ek mithai ki dukaan har Monday gajak baantti hai, aur Kabamai, jise aaj gajak mili, har 10 din mein aati hai. Use agli baar gajak kab milegi? Monday har 7 din aata hai aur wo har 10 din aati hai, isliye 7 aur 10 ka common multiple chahiye. Sabse chhota aaj se 70 din baad hai.

Idli-Vada: pehla number jis pe “idli-vada” bola jaayega. (a) 4 aur 6: 12. (b) 7 aur 11: 77. (c) 14 aur 30: 210. (d) 15 aur 55: 165.

Kya ye hamesha LCM hai? Samjhao. Haan. “Idli-vada” un numbers pe bola jaata hai jo dono ke multiple hon, aur 1 se upar ginte hue pehla aisa number lowest common multiple hi hai.

Prime factorisation se LCM (pages 56–58)

Kya har multiple ke primes mein number ke primes hote hain? Haan. Multiple, number guna koi whole number hai, isliye uske prime factorisation mein number ke apne primes aur us whole number ke primes hote hain.

Kya 2 × 3 × 5 × 7, 14 = 2 × 7 aur 35 = 5 × 7 ka common multiple hai? Haan. Isme 2 × 7 bhi hai aur 5 × 7 bhi: 210 = 14 × 15 = 35 × 6. Par ye sabse chhota nahi; 2 × 5 × 7 = 70 hai.

96 aur 360 ke liye paanch se zyaada 2 lene se sabse chhota common multiple kyun nahi milta? Har extra 2 number ko dugna karta hai. Paanch 2 mein dono numbers ke 2 aa jaate hain, isliye chhatha 2 zaroorat se dugna bada common multiple deta hai.

Figure it Out: LCM nikaalo. Har prime rakho, har ek sabse zyaada baar. (a) 30 = 2 × 3 × 5, 72 = 2 × 2 × 2 × 3 × 3: LCM = 2 × 2 × 2 × 3 × 3 × 5 = 360. (b) 36 = 2 × 2 × 3 × 3, 54 = 2 × 3 × 3 × 3: LCM = 2 × 2 × 3 × 3 × 3 = 108. (c) 105 = 3 × 5 × 7, 195 = 3 × 5 × 13, 65 = 5 × 13: LCM = 3 × 5 × 7 × 13 = 1365. (d) 222 = 2 × 3 × 37, 370 = 2 × 5 × 37: LCM = 2 × 3 × 5 × 37 = 1110.

Patterns aur properties (pages 58–60)

Aise aur jode dhoondho jinka HCF unme se ek number ho, aur unhe samjhao. Jaise 4 aur 12, 5 aur 35, 9 aur 27, 7 aur 7. Har jode mein chhota number bade ka factor hai, yaani bada number chhote ka multiple.

Agar m aisa ek number ho to doosra kya ho sakta hai? Agar 7k ho to? Agar m HCF hai, to doosra m ka koi bhi multiple hai: 2m, 3m, 5m waghairah. Agar doosra number HCF hai, to wo m ka koi factor hai. 7k ke liye: 7k ka koi multiple, jaise 14k ya 21k, ya 7k ka koi factor, jaise 7 ya k.

Figure it Out 1. HCF ke general kathan. (a) Do lagaatar even numbers: HCF 2 hai. Dono even hain, aur sirf 2 ki doori pe hain, isliye 2 se bada koi number dono ko bhaag nahi de sakta (uske multiples 2 se zyaada door hote hain). Jaise 14 aur 16. (b) Do lagaatar odd numbers: HCF 1 hai. Common factor ko unka antar, 2, bhi bhaag dena chahiye, par odd numbers 2 se nahi bantte. Jaise 15 aur 17. (c) Do even numbers: HCF even hai, kam se kam 2, kyunki 2 hamesha saajha hai. Jaise 12 aur 18 ka HCF 6. (d) Do lagaatar numbers: HCF 1 hai. 1 se bade kisi number ke multiples 1 se zyaada door hote hain, isliye wo dono ko bhaag nahi de sakta. Jaise 20 aur 21. (e) Do co-prime numbers: HCF 1 hai, kyunki co-prime ka matlab hi hai ki 1 ke alaava koi factor saajha nahi.

2. 3 aur 24 ka LCM 24 hai. (a) Aise aur jode: 4 aur 20 (LCM 20), 5 aur 15 (LCM 15), 6 aur 36 (LCM 36), 9 aur 9 (LCM 9). (b) LCM unme se ek number tab hota hai jab bada number chhote ka multiple ho. Algebra mein: n aur kn ke liye, jahan k whole number hai, LCM kn hai (aur HCF n).

3. LCM ke general kathan. (a) 3 ke do multiples: LCM 3 ka multiple hai. Jaise 6 aur 9 ka LCM 18. (b) Do lagaatar even numbers: LCM unke guna ka aadha hai, kyunki unka HCF 2 hai. Jaise 10 aur 12 ka LCM 60. (c) Do lagaatar numbers: LCM unka guna hai, kyunki HCF 1 hai. Jaise 8 aur 9 ka LCM 72. (d) Do co-prime numbers: LCM unka guna hai. Jaise 4 aur 9 ka LCM 36.

Math Talk: dono numbers dugne karne pe HCF ka kya hota hai? Wo dugna ho jaata hai. Har factorisation mein ek 2 aur judta hai, aur wo 2 saajha hai, isliye saajha hissa ek 2 badh jaata hai. Jaise HCF(6, 8) = 2 aur HCF(12, 16) = 4.

Ek hi number ke multiples ka HCF. (a) 18 × 10 aur 18 × 15: 10 aur 15 mein 5 abhi bhi saajha hai, isliye HCF = 18 × 5 = 90. (b) 10 × 38 aur 10 × 21: 38 aur 21 mein kuch saajha nahi, isliye HCF = 10. (c) 5 × 13 aur 5 × 20: 13 aur 20 mein kuch saajha nahi, isliye HCF = 5. (d) 12 × 16 aur 12 × 20: 16 aur 20 mein 4 saajha hai, isliye HCF = 12 × 4 = 48.

HCF common multiplier ke barabar kab hota hai? (b) aur (c) mein: jab baaki do multipliers co-prime hon. Agar unme kuch saajha ho, jaise 10 aur 15 ya 16 aur 20 mein, to HCF bada hota hai.

Aasaan tareeke (pages 60–62)

84 aur 180 ka ladder samjhao. Dono numbers ko ek saajha prime se bhaag dekar dono bhaagfal neeche likhe gaye: 2 se 42 aur 90, phir 2 se 21 aur 45, aur 3 se 7 aur 15. Ye ruk jaata hai kyunki 7 aur 15 mein koi prime saajha nahi.

Isse 84 aur 180 ka HCF kaise milta hai? Baayein ke divisors ka guna karo: 2 × 2 × 3 = 12. Ye theek wahi primes hain jo dono numbers mein saajha hain.

Dono ladders se HCF nikaalo. 300 aur 150: divisors 2, 5, 5 aur 3 ke baad 2 aur 1 bachte hain, isliye HCF = 2 × 5 × 5 × 3 = 150. 630 aur 770: divisors 2, 5 aur 7 ke baad 9 aur 11 bachte hain, isliye HCF = 2 × 5 × 7 = 70.

Isse LCM kaise milta hai? Divisors ko neeche bache numbers se guna karo. 300 aur 150 ke liye: 2 × 5 × 5 × 3 × 2 × 1 = 300. 630 aur 770 ke liye: 2 × 5 × 7 × 9 × 11 = 6930 (book 9 ko aage 3 × 3 mein todti hai, jawab wahi). 84 aur 180 ke liye: 12 × 7 × 15 = 1260.

Ye LCM kyun hain? Har number divisors guna apna neeche ka number hai (300 = 150 × 2 aur 150 = 150 × 1), isliye guna mein dono numbers ka poora factorisation aa jaata hai. Neeche ke numbers mein koi prime saajha nahi, isliye kuch hataaya nahi ja sakta, aur ye sabse chhota aisa number hai.

Guna aur Anshu ke shortcut kyun chalte hain? 50 = 2 × 5 × 5 se ek saath bhaag dena 2, 5 aur 5 se ek ke baad ek bhaag dene jaisa hi hai, aur 10 se bhaag dena 2 aur 5 se bhaag dena hai. Koi bhi common factor ek kadam mein nikaal sakte hain, bas tab tak chalte raho jab tak bache numbers mein kuch saajha na rahe.

Karke dekho. (a) 90 aur 150: 30 se bhaag dekar 3 aur 5, isliye HCF = 30 aur LCM = 30 × 3 × 5 = 450. (b) 84 aur 132: 12 se bhaag dekar 7 aur 11, isliye HCF = 12 aur LCM = 12 × 7 × 11 = 924.

Do numbers ka LCM bada hai ya unka guna? Kyun? LCM kabhi bada nahi hota. Guna khud ek common multiple hai (wo har number ka multiple hai), aur LCM sabse chhota common multiple hai, isliye LCM ≤ guna. Jab numbers mein 1 ke alaava kuch saajha na ho, tab dono barabar hote hain.

Kya LCM guna ka factor hai? Use kis se guna karte hain? Haan, har case mein, aur multiplier HCF hai. (a) 45 aur 105: LCM = 315, guna = 4725 = 315 × 15, aur HCF = 15. (b) 275 aur 352: LCM = 8800, guna = 96,800 = 8800 × 11, aur HCF = 11. (c) 222 aur 370: LCM = 1110, guna = 82,140 = 1110 × 74, aur HCF = 74.

HCF × LCM = do numbers ka guna kyun? Primes ko do hisson mein baanto: jo dono numbers mein saajha hain aur jo sirf ek mein hain. HCF saajha primes se banta hai. LCM saajha primes ek baar aur, saath mein baaki sab se banta hai. Isliye HCF × LCM mein saajha primes do baar aur baaki ek baar hote hain, jo theek do numbers ke guna mein hota hai. Ginti mein: agar koi prime ek number mein x baar aur doosre mein y baar aata hai, to HCF chhoti ginti leta hai aur LCM badi, aur dono milkar x + y banate hain.

Kya ye teen numbers pe chalta hai? Aam taur pe nahi. 2, 4 aur 8 ke liye: HCF = 2 aur LCM = 8, isliye HCF × LCM = 16, par guna 64 hai. Kuch khaas cases mein chal jaata hai, jaise 2, 3 aur 5 (1 × 30 = 30), par ye niyam nahi hai.

Figure it Out (pages 63–64)

1. Neele sitaare agli baar kab milenge? Upar ki kataar mein rang har 6 sitaaron pe dohraate hain, aur neela 4, 10, 16, … jagah pe hai. Neeche ki kataar mein har 4 sitaaron pe, aur neela 4, 8, 12, 16, … pe. Neele sitaare jagah 4 pe milte hain, aur agli baar jagah 16 pe, 12 jagah baad, kyunki 6 aur 4 ka LCM 12 hai. Uske baad har 12 sitaaron pe milte hain: 28, 40 aur aage.

2. (a) Kya 5 × 7 × 11 × 11, 5 × 7 × 7 × 11 × 2 ka multiple hai? Nahi. Multiple mein 5 × 7 × 7 × 11 × 2 ke saare primes hone chahiye, par 5 × 7 × 11 × 11 mein koi 2 nahi aur sirf ek 7 hai. (b) Kya ye uska factor hai? Nahi. Factor 5 × 7 × 7 × 11 × 2 ka hissa hona chahiye, par 5 × 7 × 11 × 11 mein do 11 hain aur us number mein sirf ek.

3. HCF aur LCM prime factorisation ke roop mein. (a) 3 × 3 × 5 × 7 × 7 aur 12 × 7 × 11 = 2 × 2 × 3 × 7 × 11: HCF = 3 × 7 (= 21); LCM = 2 × 2 × 3 × 3 × 5 × 7 × 7 × 11 (= 97,020). (b) 45 = 3 × 3 × 5 aur 36 = 2 × 2 × 3 × 3: HCF = 3 × 3 (= 9); LCM = 2 × 2 × 3 × 3 × 5 (= 180).

4. Do numbers jinka HCF 1 aur LCM 66 ho. 66 = 2 × 3 × 11. Iske primes do numbers mein aise baanto ki kuch saajha na ho: 6 aur 11 chalte hain, aur 2 aur 33, 3 aur 22, aur 1 aur 66 bhi.

5. Gwaale ki gaayein (200 se kam). 3, 5 aur 7 darwaazon se barabar gaayein guzrin, isliye gaayon ki sankhya 3, 5 aur 7 ka multiple hai, yaani unke LCM 3 × 5 × 7 = 105 ka multiple. 200 se kam aisa ek hi number hai: 105 gaayein.

6. 12 cm × 18 cm × 36 cm ke dibbe mein bina jagah chhode kaunse cubes bharenge? Side 12, 18 aur 36 teeno ko bhaag de, yaani common factor ho: 1, 2, 3 ya 6 cm. Isliye (b) 6 cm, (d) 3 cm aur (e) 2 cm fit hote hain. 9 cm, 12 ko nahi bhaag deta, aur 4 cm, 18 ko nahi.

7. Diye gaye numbers mein sabse bada jo 306 aur 36 dono ko bhaag de. 306 = 2 × 3 × 3 × 17 aur 36 = 2 × 2 × 3 × 3, isliye HCF 2 × 3 × 3 = 18. Jawab: (c) 18. (36, 306 ko nahi bhaag deta, aur 612 aur 360, 36 se bade hain.)

8. 3, 4, 5 aur 7 se bantne waala sabse chhota number jo 11 se bhaag dene pe 10 bachaaye. Ye 3, 4, 5 aur 7 ke LCM, 420, ka multiple hona chahiye. Multiples jaancho: 420, 840, 1260 aur 1680, 11 se bhaag dene pe 2, 4, 6 aur 8 bachaate hain, aur 2100, 10 bachaata hai (11 × 190 = 2090). Jawab: 2100.

9. ‘Fire in the Mountain’: 6 ya 9 bolne pe koi out nahi hua, par 10 bolne pe kuch out hue. Bachchon ki sankhya 6 aur 9 ke groups mein poori bant jaati hai, isliye wo 6 aur 9 ke LCM, 18, ka multiple hai, par 10 ka multiple nahi. Vikalpon mein (a) 72 aur (e) 36 sahi baithte hain. 90, 10 ka multiple hai, isliye koi out nahi hota; 45 aur 3, 6 ke multiple nahi hain.

10. Do alag primes m aur n ka LCM ho sakta hai: Primes mein kuch saajha nahi, isliye unka LCM m × n hai, jo dono se bada hai. Sirf (c) dono numbers se bada sahi hai. Ye dono se chhota ya beech mein nahi ho sakta, aur m × n se na chhota hai na bada, kyunki m × n ke barabar hai.

11. Kutta 150 ft aage khade khargosh ka peechha karta hai; har baar khargosh 7 ft koodta hai to kutta 9 ft. Kitni chhalaangon mein pakdega? Har chhalaang mein kutta 9 − 7 = 2 ft paas aata hai, isliye 150 ÷ 2 = 75 chhalaang. Jaanch: kutta 75 × 9 = 675 ft chalta hai aur khargosh 150 + 75 × 7 = 675 ft pe hai.

12. 1, 2, 3, 4, 5, 6, 8, 9 aur 10 ka sabse chhota multiple. Isme teen 2 (8 ke liye), do 3 (9 ke liye) aur ek 5 chahiye: 2 × 2 × 2 × 3 × 3 × 5 = 360. Ye Grade 6 ke jawab se milta hai, 1 se 10 tak bina 7 ke, jo Class 6 Prime Time ke solutions mein hai.

13. Mahaviracharya ka jod: 8/15 + 1/20 + 7/36 + 11/63 + 1/21. Denominators ka LCM common denominator banao. 15 = 3 × 5, 20 = 2 × 2 × 5, 36 = 2 × 2 × 3 × 3, 63 = 3 × 3 × 7 aur 21 = 3 × 7, isliye LCM 2 × 2 × 3 × 3 × 5 × 7 = 1260. Phir 8/15 = 672/1260, 1/20 = 63/1260, 7/36 = 245/1260, 11/63 = 220/1260 aur 1/21 = 60/1260. Jod (672 + 63 + 245 + 220 + 60)/1260 = 1260/1260 = 1 hai.

Summary box aur Mystery Colours (pages 65–66)

Ab tak ke sabse bade prime mein 4,10,24,320 digits hain. Ise likhne mein kitna samay lagega? Ye 41,024,320 digits hain. Bina ruke har second ek digit likhne pe 41,024,320 second, yaani lagbhag 475 din lagenge. Roz 8 ghante likhne pe lagbhag 1425 din, yaani kareeb 4 saal. (Extra: ye ek Mersenne prime hai, 2 ko khud se 136,279,841 baar guna karne se 1 kam.)

Page numbers ke chaaron taraf ki rings ka colour scheme samjho. Kisi number ki ring utne barabar hisson mein bantti hai jitne uske prime factors hain, dohraane waale bhi gin kar, aur har hissa apne prime ka rang leta hai: 2 hara, 3 baingani, 5 naarangi, 7 neela, aur 11 ya usse bada lal, jahan lal hisse ke paas prime chhota likha hota hai. 1 ka koi prime factor nahi, isliye uski ring grey hai, aur har prime ki poori ring ek hi rang ki hai. Patterns: even numbers mein hamesha hara hissa, 3 ke multiples mein baingani hissa, 2 ki powers poori hari, aur hisson ki ginti primes ki ginti se badhti hai, number ke size se nahi.

Page numbers 101 se 110 rango. 101, 103, 107 aur 109 primes hain: poori lal rings. 102 = 2 × 3 × 17: hara, baingani, lal. 104 = 2 × 2 × 2 × 13: teen hare hisse aur ek lal. 105 = 3 × 5 × 7: baingani, naarangi, neela. 106 = 2 × 53: hara aur lal. 108 = 2 × 2 × 3 × 3 × 3: do hare aur teen baingani hisse. 110 = 2 × 5 × 11: hara, naarangi, lal.