✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- Cryptarithm mein ek letter = ?
- Pehla digit 0 ho sakta?
- 13 × 8 kitne digit ka?
Answers dekho
- Ek hi digit (alag letter = alag digit)
- Nahi
- 3 digit (104)
Teeno sahi? Badhiya, aage chaliye! Koi galat hua? Koi baat nahi, Post 5 — Divisibility Rule: 9, 3 Aur 11 aur Post 6 — Divisibility: 4, 8 Aur Digital Root ke rules dekh lijiye.
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🎬 Kahani: Jasoos Jiya Ka Secret Code
Jiya ko ek chit mili: AB + AB + AB = CAB. Neeche likha tha: “Har letter ek digit, alag letter alag digit. Code kholo, treasure milega!” 🗝️
Uske dost ne turant guess shuru kiye: 12? 12 × 3 = 36… 23? … 41? Aadha ghanta, kuch nahi.
Jiya muskurayi: “Guess nahi, tark. Teen baar AB jodo aur aakhir mein phir AB hi aaye: iska kya matlab?” Do minute mein code khul gaya. Aaj woh tareeka seekhte hain! 🕵️♀️
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📖 Concept: Guess Nahi, Shartein Ghataao
💡 Is post ki ek badi baat: Cryptarithm guess se nahi, possibilities kaatne se khulta: units digit, carry, number ka size (kitne digit), aur divisibility. Har tark kuch digits ko bahar karta, jab tak ek hi jawab bache. Phir wapas daal ke check.
Step 1: Hathiyaar 1: units digit
AB + AB + AB = CAB: aakhir mein B. 3 × B ka units digit B → 2B ka units 0 → B = 0 ya 5.
WHY? Jodne mein units digit sirf units digits se banta. Baaki digits wahan nahi pahunchte.
Step 2: Hathiyaar 2: poora number dekho
3 × AB = 100C + AB → 2 × AB = 100C → AB = 50C. AB 2-digit → C = 1 → AB = 50.
Check: 50 + 50 + 50 = 150 = CAB (C = 1, A = 5, B = 0) ✅ Jiya ka treasure!
Step 3: Hathiyaar 3: size (kitne digit)
A1 × A = 1BA: jawab 100 aur 199 ke beech. A = 3: 31 × 3 = 93 (chhota). A = 5: 51 × 5 = 255 (bada). A = 4: 41 × 4 = 164 ✅
Units bhi check: 1 × A = A ✅ (units A). B = 6.
WHY size? Digits ki ginti se range pakki ho jaati, aur guess 2-3 pe aa jaate.
Step 4: Hathiyaar 4: carry
TOO + TOO = FOUR: 3-digit + 3-digit = 4-digit → carry → F = 1, aur T ≥ 5.
Units: O + O ka units R; tens: O + O + carry ka units U; hundreds: T + T + carry = 10 + O.
Try O = 7: R = 4, carry 1; tens: 7 + 7 + 1 = 15 → U = 5, carry 1; T + T + 1 = 17 → T = 8. 877 + 877 = 1754 ✅ (Doosre O se koi alag digit nahi bante; try karo!)
Step 5: Hathiyaar 5: divisibility (missing digit)
31z5 jaisa: 9 se bate → digit jod (Post 5 — Divisibility Rule: 9, 3 Aur 11). 6 se bate → even aur 3 se (Post 6 — Divisibility: 4, 8 Aur Digital Root).
Udaharan: 4a7b, 5 se bhi aur 9 se bhi → b = 0 ya 5. b = 0: 11 + a → a = 7. b = 5: 16 + a → a = 2. Do jawab: 4770, 4275.
Kab kaunsa hathiyaar? Jod/guna ke cryptarithm: pehle units, phir size/carry. Missing digit: divisibility. Algebra (AB = 10A + B) jab chhota-mota tark kaam na aaye.
🪜 Atak Gaye? Aise Socho
- Shartein likho: alag letter alag digit; pehla digit 0 nahi.
- Units digit se shuru karo: kaun se digits possible?
- Kitne digit ka jawab? Range nikaalo.
- Carry dekho: kya agla column 1 lega?
- Bacha hua 2-3 options mein try karo, aur jawab daal ke check karo.
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🔐 Deep Concept: Tark Ki Chhalni: Har Shart Ek Filter
Socho 10 digits ek chhalni (sieve) pe. Har shart ek chhalni: units digit wali, size wali, carry wali. Har chhalni ke baad kam digits bachte. Aakhri chhalni ke baad sirf ek! 🧺
Algebra wali nazar: AB × 9 = CBA ko likho: 9(10A + B) = 100C + 10B + A → 89A − B = 100C. B ek digit (0–9), to 89A ka aakhri do digit 09 ya kam chahiye. Sirf A = 9: 801 = 800 + 1 → C = 8, B = 1. 91 × 9 = 819 ✅
Ek aur: AB × 7 = CAB → 6 × AB = 100C → AB = 50C/3 → C = 3, AB = 50: 50 × 7 = 350 ✅. Jiya wale code jaisa hi pattern!
Unifying idea: poore chapter (parity, remainder, divisibility) ke tools yahan ek saath kaam aate. Cryptarithm = number sense ka final exam. 🎯
sirf charumam.com readers ke liye 🔐
✏️ Solved Examples
Example 1 (Easy): AA + BB + CC = ABC
AA = 11A, to 11(A + B + C) = ABC → ABC 11 ka multiple. A + B + C ≤ 24, aur ABC ≥ 100 → A = 1. Try: 11 × (1 + B + C) = 100 + 10B + C → 11 + 11B + 11C = 100 + 10B + C → B + 10C = 89 → C = 8, B = 9. 11 + 99 + 88 = 198.
Check: 11 + 99 + 88 = 198 ✅, A = 1, B = 9, C = 8 alag ✅
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Example 2 (Medium): AB × AB = CAB
AB² ka aakhri do digit AB. Units: B² ka units B → B = 0, 1, 5, 6. Try 25: 625 ✅ (76² = 5776, 4-digit ❌).
Check: 25 × 25 = 625: C = 6, A = 2, B = 5 ✅
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Example 3 (NCERT style): 73a5b 18 se bate. (a, b)?
18 = 2 × 9 (co-prime). b even; 7 + 3 + a + 5 + b = 15 + a + b = 18 ya 27 → a + b = 3 ya 12. Jodiyaan: (3, 0), (1, 2), (8, 4), (6, 6), (4, 8).
Check: 73854 = 18 × 4103 ✅, 73350 = 18 × 4075 ✅
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Jiya ka code: 50 + 50 + 50 = 150. Treasure box ka taala 150 pe khula, andar ek chit: “Shabaash, jasoos! Agla code: ABC × 4 = DCBA.” (Jawab: 802 × 4 = 3208. Try karo!) 🎁
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🏠 Cryptarithm Wali Soch Roz Ki Zindagi Mein
- 🔒 Lock code: “aakhri digit even, jod 9 ka multiple”: shartein ghataa ke code kam options mein.
- 🕵️ Jasoosi: Sherlock Holmes jaisi soch: “jo namumkin hai use hata do, jo bacha wahi sach.”
- 🧩 Sudoku: wahi tark: har shart kuch digits ko bahar karti.
- 💻 Password security: agar hint zyada mile to options kam, isliye strong password mein patterns nahi.
- 🔐 Codes ka itihaas: World War mein code todne wale isi “possibilities ghataao” tark se kaam karte the.
Sudoku aur cryptarithm bhai-bhai: dono mein guess se zyada tark jeet-ta hai!
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🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| Jod wala | ✅ Units + carry | Phir size |
| Guna wala | ✅ Size se range | Phir units |
| Missing digit | ✅ Divisibility | Do jawab ho sakte |
| Aakhir mein same number | ✅ Algebra (AB = 10A + B) | Chhota equation |
| Jawab mila | ✅ Wapas daal ke check | Alag digits? |
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❌→✅ Worked Mistake
Sawaal: PQ × 8 = RS (2-digit × 8 = 2-digit, sab alag).
❌ Galat: “10 × 8 = 80, ho gaya!” Par P = 1, Q = 0, R = 8, S = 0 → Q aur S dono 0: alag letter, same digit ❌
✅ Sahi: size: PQ × 8 < 100 → PQ ≤ 12. 11 (P = Q ❌), 10 ❌ → 12 × 8 = 96. Check: 1, 2, 9, 6 sab alag ✅
Doosri common galti: pehla mila jawab pe ruk jaana. Missing digit mein aksar do jawab hote (2z43 → 0 ya 9).
🧠 Rule of thumb: Pehle shartein, phir options; jawab mile to saari shartein dobara check.
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📝 10 MCQs
Q1. AB + AB + AB = CAB mein B?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 3B ka units B → 2B ka 0 ✅
- (b) galat: B = 0 bhi possible hai.
- (c) galat: 3 × 1 = 3, units 1 nahi.
- (d) galat: Units digit se kaafi kat jaate.
Q2. 3-digit + 3-digit = 4-digit. Pehla digit?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Do 3-digit ka jod 1998 tak → 1 ✅
- (b) galat: 2000 tak nahi pahunchta.
- (c) galat: 9 ke liye bahut bada jod chahiye.
- (d) galat: Pehla digit 0 nahi hota.
Q3. PQ × 8 = RS (2-digit): PQ ki range?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 13 × 8 = 104, to 12 tak ✅
- (b) galat: 13 se upar 3-digit ban jaata.
- (c) galat: Bahut bada range, size dekho.
- (d) galat: 13 × 8 = 104 → 3-digit, bahar.
Q4. B² ka units B: B kya ho sakta?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 0, 1, 25, 36 sab mein ✅
- (b) galat: 5 akela nahi, aur bhi hain.
- (c) galat: 4 aur 16: units B nahi.
- (d) galat: 9 aur 49: units B nahi.
Q5. A1 × A = 1BA. A?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 41 × 4 = 164, 1 se shuru ✅
- (b) galat: 31 × 3 = 93, 2-digit hi.
- (c) galat: 51 × 5 = 255, 1 se shuru nahi.
- (d) galat: 21 × 2 = 42, 2-digit hi.
Q6. 2z43 9 se bate. z?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 9 + z = 9 ya 18 chahiye ✅
- (b) galat: 9 bhi chalega: 2943 = 9 × 327.
- (c) galat: 0 bhi chalega: 2043 = 9 × 227.
- (d) galat: 3 se jod 12, 9 se nahi.
Q7. AB × 7 = CAB. AB?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 6 × AB = 100C → 50, C = 3 ✅
- (b) galat: 25 × 7 = 175, 25 nahi aakhir mein.
- (c) galat: 14 × 7 = 98, 2-digit hi.
- (d) galat: 70 × 7 = 490, A = 7 nahi bacha.
Q8. 11 + 99 + 88 = 198 kis paheli ka jawab?
Solution dekho
Sahi jawab: (a)
- (a) sahi: A = 1, B = 9, C = 8 bilkul ✅
- (b) galat: AB + BA do 2-digit ka jod.
- (c) galat: Yeh guna wali paheli hai.
- (d) galat: Yeh bhi guna wali paheli.
Q9. 4a7b 5 aur 9 se bate: kitne jawab?
Solution dekho
Sahi jawab: (b)
- (a) galat: Do milte: b = 0 aur b = 5.
- (b) sahi: 4770 aur 4275, do jawab ✅
- (c) galat: Teesra koi a nahi banta.
- (d) galat: Jawab milte hain, zero nahi.
Q10. Pehla mila jawab sahi hai to?
Solution dekho
Sahi jawab: (b)
- (a) galat: Do jawab bhi ho sakte hain.
- (b) sahi: Shayad aur bhi jawab ho ✅
- (c) galat: Tark se kaam karo, guess nahi.
- (d) galat: Kaam poora karo, chhodo nahi.
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Ek letter do jagah ho to?
Dono jagah same digit. Yahi shart paheli ko kasti hai.
2. Alag letter same digit ho sakte?
Nahi: alag letter = alag digit. Isliye 10 × 8 = 80 PQ × 8 = RS ka jawab nahi.
3. Kya har cryptarithm ka ek hi jawab?
Achhi paheli mein haan, par kuch mein kai (jaise NO + NO = YES ke 16!).
4. Shuru kahan se karein?
Units digit aur sabse bade digit (carry/size) se. Wahan sabse zyada jaankari.
5. Algebra kab?
Jab number khud jawab mein dobara aaye (CAB mein AB): 10A + B likh ke chhota equation.
6. Aur paheliyan kahan?
SEND + MORE = MONEY sabse mashhoor hai: try karo (9567 + 1085 = 10652).
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy A + A + A = BA (A, B digits)?
Solution dekho
3A = 10B + A → 2A = 10B → A = 5, B = 1: 5 + 5 + 5 = 15. Check ✅
Q2. Easy AB × 6 = CAB?
Solution dekho
5 × AB = 100C → AB = 20C; A ≠ B → 20 × 6 = 120, 40 × 6 = 240, 60 × 6 = 360, 80 × 6 = 480 (sab sahi). Check: 120 → C = 1, A = 2, B = 0 ✅
Q3. Easy 5z4 3 se bate: z?
Solution dekho
9 + z → 0, 3, 6, 9. Check: 534 = 3 × 178 ✅
Q4. Easy AB + 5 = BA (A, B alag)?
Solution dekho
9(B − A) = 5? namumkin → koi jawab nahi. Check: farak hamesha 9 ka multiple ✅
Q5. Medium AB × 9 = CBA?
Solution dekho
91 × 9 = 819. Check ✅
Q6. Medium AB − BA = 9 ka koi bhi ek jawab?
Solution dekho
A − B = 1: 21 − 12 = 9 (aur 32 − 23…). Check ✅
Q7. Medium 6a5b 4 aur 9 se bate?
Solution dekho
5b ÷ 4 → b = 2, 6; jod 11 + a + b: b = 2 → a = 5; b = 6 → a = 1. 6552, 6156. Check: 6552 = 36 × 182 ✅
Q8. Medium A1 × A = 1BA ka jawab check karo: kya A = 4 akela?
Solution dekho
A = 4 → 164 ✅; A = 3 → 93 ❌; A = 5 → 255 ❌ → haan, akela. Check ✅
Q9. NCERT style ABC × 4 = DCBA?
Solution dekho
4-digit jawab → ABC ≥ 250. A = 4C ka units → A even: 4, 6 ya 8. A = 8: C = 2 ya 7, aur D = 3; try 8B2 × 4 = 3?08 → B = 0: 802 × 4 = 3208. Check: D = 3, C = 2, B = 0, A = 8 ✅
Q10. NCERT style TOO + TOO = FOUR: kya doosra jawab?
Solution dekho
Nahi, sirf 877 + 877 = 1754. Check: O ke baaki digits try karke alag digit nahi bante ✅
Q11. NCERT style 5p3q6 44 se bate: (p, q)?
Solution dekho
44 = 4 × 11. 4: aakhri do q6 → q odd (16, 36, 56…). 11: units se 6 − q + 3 − p + 5 = 14 − p − q = 0 ya 11 → p + q = 14 ya 3. Jodiyaan: (0, 3), (2, 1), (5, 9), (7, 7), (9, 5). Check: 50336 = 44 × 1144 ✅
Q12. NCERT style AB × AB = CAB ka doosra possible jawab 3-digit mein?
Solution dekho
Nahi: 76² = 5776 4-digit → sirf 625. Check ✅
Answer Key — charumam.com exclusive
🧠 HOTS (Sochne Wale Sawaal)
Q1. AB + BA ka jawab kab 3-digit?
Solution dekho
11(A + B) ≥ 100 → A + B ≥ 10. Check: 56 + 65 = 121 ✅
Q2. ABCD × 4 = DCBA?
Solution dekho
2178 × 4 = 8712. Check ✅
Q3. WOW × 5 = MEOW?
Solution dekho
5 × W ka units W → W = 0 ya 5; pehla digit → W = 5. 5O5 × 5 = 25xx–29xx → M = 2; aakhri teen digit O5 se mile → O = 7: 575 × 5 = 2875. Check: M = 2, E = 8, O = 7, W = 5 ✅
Q4. JK × 6 = KKK jaisi paheli JK × 3 = KKK?
Solution dekho
KKK = 111K = 3 × 37 × K → JK = 37K: K = 1 → 37 (K = 7 ≠ 1 ❌), K = 2 → 74 (K = 4 ≠ 2 ❌) → koi jawab nahi. Check ✅
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🏆 Olympiad Corner
Q1. SEND + MORE = MONEY mein M?
Solution dekho
1.
Q2. AB × B = CCC (alag digits)?
Solution dekho
Koi jawab nahi (37 × 7 = 259, 74 × 4 = 296…).
Q3. 6-digit abcabc kis bade prime ka multiple hamesha?
Solution dekho
13 (aur 7, 11).
Q4. NO + NO = YES ke kitne jawab?
Solution dekho
16.
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💗 CharuMam Motivation
Jiya jasoos isliye jeeti kyunki usne guess nahi kiye, shartein ginee. Poore chapter ki saari tricks ek paheli mein! Aap bhi ab number-jasoos ho. 🕵️
Chapter 5 poora! Lagaataar jod se cryptarithm tak, har jagah ek hi sawaal: “yeh kyun kaam karta hai?”
— 📌 charumam.com
🟢 Master Recap
- 💡 Chaabi: Cryptarithm guess se nahi, possibilities kaatne se khulta: units digit, carry, number ka size (kitne digit), aur divisibility. Har tark kuch digits ko bahar karta, jab tak ek hi jawab bache. Phir wapas daal ke check.
- Alag letter = alag digit; pehla digit 0 nahi.
- Units digit se shuru.
- Size (kitne digit) se range.
- Carry: 3-digit + 3-digit → 4-digit to 1.
- Missing digit: divisibility rules.
- Jawab daal ke check; doosre jawab bhi dhundho.
— charumam.com 🎯
👉 Aage Kya?
Ab numbers se algebra ki taraf: guna kaise “baant-ta” hai aur (a + b)² ka raaz: Chapter 6: We Distribute, Yet Things Multiply ka hub dekhiye. ✖️
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬