Dosto, Maths ka safar yahin se shuru hota hai! Ye chapter numbers ke types (natural, whole, prime…), place value, divisibility rules, BODMAS aur simplification ke shortcuts cover karta hai.
HSSC CET Group D mein Maths ke 10 questions aate hain, aur unme se 2–3 questions simplification / number system se lagbhag pakke hote hain. Ye sabse aasan marks hain — bas rules yaad hone chahiye.
Tension mat lo — har concept chhote steps mein samjhaya hai, solved examples ke saath. End mein 14 MCQs ka quiz hai. Chalo shuru karein! 💪
1. Numbers ke Types (Prakar)
| Type | Kya hota hai | Examples |
|---|---|---|
| Natural (Prakritik) | Ginti ke numbers — 1 se shuru | 1, 2, 3, 4, … |
| Whole (Purn) | Natural + 0 | 0, 1, 2, 3, … |
| Integers (Purnank) | Whole + negative numbers | …, −2, −1, 0, 1, 2, … |
| Even (Sam) | 2 se poora divide ho | 2, 4, 6, 8, … |
| Odd (Visham) | 2 se divide na ho | 1, 3, 5, 7, … |
| Prime (Abhajya) | Sirf 2 factors — 1 aur khud | 2, 3, 5, 7, 11, 13, … |
| Composite (Bhajya) | 2 se zyada factors | 4, 6, 8, 9, 10, … |
| Co-prime (Sah-abhajya) | Do numbers jinka HCF = 1 | (4, 9), (8, 15) |
📌 Exam ke 4 GOLDEN facts
① 1 na prime hai na composite ② 2 aklauta EVEN prime hai ③ 0 whole number hai par natural nahi ④ Sabse chhota composite number = 4
1 se 100 tak total 25 prime numbers hain (1 se 50 tak 15): 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 (yahan tak 15) aur aage 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
2. Place Value vs Face Value
- Face value (Ankit maan) = digit ki apni value. Kisi bhi jagah ho, same rehti hai.
- Place value (Sthaniya maan) = digit × uski jagah (place) ki value.
✏️ Solved Example
Number 25,634 mein 5 ki:
• Face value = 5 (bas digit khud)
• Place value = 5 × 1000 = 5000 (kyunki 5 thousands wali jagah par hai)
• Face value = 5 (bas digit khud)
• Place value = 5 × 1000 = 5000 (kyunki 5 thousands wali jagah par hai)
3. Divisibility Rules (Vibhajyata ke Niyam)
| Number | Rule (kab divide hoga) | Example |
|---|---|---|
| 2 | Last digit even ho (0,2,4,6,8) | 128 ✓ |
| 3 | Digits ka SUM 3 se divide ho | 456 → 4+5+6=15 ✓ |
| 4 | Last 2 digits 4 se divide hon | 716 → 16÷4 ✓ |
| 5 | Last digit 0 ya 5 ho | 435 ✓ |
| 6 | 2 AUR 3 dono se divide ho | 132 ✓ |
| 8 | Last 3 digits 8 se divide hon | 5104 → 104÷8 ✓ |
| 9 | Digits ka SUM 9 se divide ho | 45783 → 27 ✓ |
| 10 | Last digit 0 ho | 340 ✓ |
| 11 | (Odd places ka sum) − (Even places ka sum) = 0 ya 11 ka multiple | 245157 → 0 ✓ |
✏️ Solved Example — 11 ka rule
Kya 245157, 11 se divide hota hai?
Step 1: Odd places (right se 1st, 3rd, 5th): 7 + 1 + 4 = 12
Step 2: Even places (2nd, 4th, 6th): 5 + 5 + 2 = 12
Step 3: Difference = 12 − 12 = 0 → Haan, divide hota hai! ✓
Step 1: Odd places (right se 1st, 3rd, 5th): 7 + 1 + 4 = 12
Step 2: Even places (2nd, 4th, 6th): 5 + 5 + 2 = 12
Step 3: Difference = 12 − 12 = 0 → Haan, divide hota hai! ✓
4. BODMAS — Simplification ka MASTER RULE
Jab ek hi sawal mein +, −, ×, ÷, brackets sab ho, to solve karne ka fix order:
| Order | Letter | Matlab |
|---|---|---|
| 1 | B — Bracket | Pehle bracket: ( ) → { } → [ ]. Bar (line) sabse pehle! |
| 2 | O — Of | 'of' matlab multiply (× se pehle hota hai!) |
| 3 | D — Division (÷) | ÷ aur × barabar hain — LEFT se RIGHT jo pehle aaye |
| 4 | M — Multiplication (×) | |
| 5 | A — Addition (+) | + aur − bhi barabar — LEFT se RIGHT |
| 6 | S — Subtraction (−) |
✏️ Solved Example 1 (basic)
12 + 8 ÷ 4 × 3 − 5 = ?
Step 1: Division pehle (left se): 8 ÷ 4 = 2 → 12 + 2 × 3 − 5
Step 2: Multiplication: 2 × 3 = 6 → 12 + 6 − 5
Step 3: Left se right: 12 + 6 = 18, phir 18 − 5 = 13 ✓
Step 1: Division pehle (left se): 8 ÷ 4 = 2 → 12 + 2 × 3 − 5
Step 2: Multiplication: 2 × 3 = 6 → 12 + 6 − 5
Step 3: Left se right: 12 + 6 = 18, phir 18 − 5 = 13 ✓
✏️ Solved Example 2 (brackets)
25 − [8 − {6 − (4 − 2)}] = ?
Step 1: Sabse andar wala ( ): 4 − 2 = 2 → 25 − [8 − {6 − 2}]
Step 2: Curly { }: 6 − 2 = 4 → 25 − [8 − 4]
Step 3: Square [ ]: 8 − 4 = 4 → 25 − 4 = 21 ✓
Step 1: Sabse andar wala ( ): 4 − 2 = 2 → 25 − [8 − {6 − 2}]
Step 2: Curly { }: 6 − 2 = 4 → 25 − [8 − 4]
Step 3: Square [ ]: 8 − 4 = 4 → 25 − 4 = 21 ✓
✏️ Solved Example 3 ('of' ka khel — TRAP!)
18 ÷ 3 of 2 = ?
Step 1: 'Of' pehle solve hota hai (÷ se bhi pehle): 3 of 2 = 3 × 2 = 6
Step 2: 18 ÷ 6 = 3 ✓
⚠️ Agar 'of' ki jagah × hota (18 ÷ 3 × 2), to left-se-right: 6 × 2 = 12. Dono alag hain!
Step 1: 'Of' pehle solve hota hai (÷ se bhi pehle): 3 of 2 = 3 × 2 = 6
Step 2: 18 ÷ 6 = 3 ✓
⚠️ Agar 'of' ki jagah × hota (18 ÷ 3 × 2), to left-se-right: 6 × 2 = 12. Dono alag hain!
5. Kaam ke Sum Formulas
| Series | Formula | Example |
|---|---|---|
| 1 + 2 + 3 + … + n | n(n+1)/2 | 1 se 50 tak = 50×51/2 = 1275 |
| 1² + 2² + … + n² | n(n+1)(2n+1)/6 | 1 se 10 tak = 10×11×21/6 = 385 |
| 1³ + 2³ + … + n³ | [n(n+1)/2]² | 1 se 5 tak = (15)² = 225 |
| Pehle n ODD numbers | n² | Pehle 20 odd = 20² = 400 |
| Pehle n EVEN numbers | n(n+1) | Pehle 15 even = 15×16 = 240 |
6. Unit Digit (Ikai ka Ank) Nikalna
Bade powers ka unit digit nikalne ke liye cycle yaad karo:
| Digit | Cycle | Cycle length |
|---|---|---|
| 0, 1, 5, 6 | Hamesha wahi digit | 1 |
| 4 | 4, 6, 4, 6… | 2 |
| 9 | 9, 1, 9, 1… | 2 |
| 2 | 2, 4, 8, 6 | 4 |
| 3 | 3, 9, 7, 1 | 4 |
| 7 | 7, 9, 3, 1 | 4 |
| 8 | 8, 4, 2, 6 | 4 |
✏️ Solved Example — 745 ka unit digit
Step 1: 7 ka cycle = 7, 9, 3, 1 (length 4)
Step 2: Power ko 4 se divide karo: 45 ÷ 4 = 11, remainder 1
Step 3: Remainder 1 → cycle ka 1st digit = 7 ✓ (remainder 0 hota to cycle ka LAST digit lete)
Step 2: Power ko 4 se divide karo: 45 ÷ 4 = 11, remainder 1
Step 3: Remainder 1 → cycle ka 1st digit = 7 ✓ (remainder 0 hota to cycle ka LAST digit lete)
Trick 1: BODMAS yaad rakhne ka order
B O D M A S
Bracket → Of → Divide → Multiply → Add → Subtract. Brackets ke andar order: ( ) pehle, phir { }, phir [ ] — "chhota bracket, pehla number!"
B O D M A S
Bracket → Of → Divide → Multiply → Add → Subtract. Brackets ke andar order: ( ) pehle, phir { }, phir [ ] — "chhota bracket, pehla number!"
Trick 2: 5 par khatam hone wale number ka SQUARE
n5² = n×(n+1) | 25
35² → 3×4 = 12, aage 25 laga do = 1225. 75² → 7×8 = 56 → 5625. 2 second ka kaam!
n5² = n×(n+1) | 25
35² → 3×4 = 12, aage 25 laga do = 1225. 75² → 7×8 = 56 → 5625. 2 second ka kaam!
Trick 3: 99, 999 se multiply
×999 = ×1000 − ×1
47 × 999 = 47000 − 47 = 46953. Waise hi 47 × 99 = 4700 − 47 = 4653. Kabhi seedha multiply mat karo!
×999 = ×1000 − ×1
47 × 999 = 47000 − 47 = 46953. Waise hi 47 × 99 = 4700 − 47 = 4653. Kabhi seedha multiply mat karo!
Trick 4: ×5, ×25 ka jugaad
×5 = ×10÷2 · ×25 = ×100÷4
84 × 5 = 840 ÷ 2 = 420. 36 × 25 = 3600 ÷ 4 = 900. Zero laga ke divide — bas!
×5 = ×10÷2 · ×25 = ×100÷4
84 × 5 = 840 ÷ 2 = 420. 36 × 25 = 3600 ÷ 4 = 900. Zero laga ke divide — bas!
Trick 5: Prime check (100 tak)
2, 3, 5, 7 se test
100 tak ke kisi bhi number ko sirf 2, 3, 5, 7 se divide karke dekho — kisi se divide nahi hua to prime hai. Jaise 91 = 7 × 13, isliye 91 prime NAHI (famous trap!).
2, 3, 5, 7 se test
100 tak ke kisi bhi number ko sirf 2, 3, 5, 7 se divide karke dekho — kisi se divide nahi hua to prime hai. Jaise 91 = 7 × 13, isliye 91 prime NAHI (famous trap!).
❌ Galti 1: DM ka matlab "pehle Divide, hamesha"… NAHI!
÷ aur × ka rank BARABAR hai — jo LEFT mein pehle aaye wahi pehle hoga. Yahan multiply LEFT mein hai, isliye multiply pehle:
12 × 8 ÷ 4 × 2 = ?
✅ Sahi (left se right): 12 × 8 = 96 → 96 ÷ 4 = 24 → 24 × 2 = 48
❌ Galat (÷ chhod ke aage wala × pehle kiya): 4 × 2 = 8 → 96 ÷ 8 = 12
Bilkul yahi rule + aur − par bhi: 20 − 8 + 3 = 15 (na ki 20 − 11 = 9).
12 × 8 ÷ 4 × 2 = ?
✅ Sahi (left se right): 12 × 8 = 96 → 96 ÷ 4 = 24 → 24 × 2 = 48
❌ Galat (÷ chhod ke aage wala × pehle kiya): 4 × 2 = 8 → 96 ÷ 8 = 12
Bilkul yahi rule + aur − par bhi: 20 − 8 + 3 = 15 (na ki 20 − 11 = 9).
❌ Galti 2: Minus ke baad bracket kholne mein sign bhoolna
10 − (4 − 6) = 10 − (−2) = 12. Bracket ke aage minus ho to andar ke SAB signs ulte ho jaate hain.
❌ Galti 3: 'of' ko normal × samajhna
'Of' ki priority ÷ se ZYADA hai. 18 ÷ 3 of 2 = 18 ÷ 6 = 3, jabki 18 ÷ 3 × 2 = 12.
❌ Galti 4: "1 prime hai" likh dena
1 ke paas sirf EK factor hai (khud), isliye wo na prime na composite. Aur "sabse chhota prime" = 2 (na ki 1 ya 3).
❌ Galti 5: Place value aur Face value mix karna
25,634 mein 5 ki face value 5 hai, place value 5000. Question dhyan se padho — dono alag cheez hain!
1 + 2 + … + n = ?tap 👆
n(n+1)/2
1² + 2² + … + n² = ?tap 👆
n(n+1)(2n+1)/6
1³ + 2³ + … + n³ = ?tap 👆
[n(n+1)/2]²
Pehle n ODD numbers ka sum?tap 👆
n²
Pehle n EVEN numbers ka sum?tap 👆
n(n+1)
Sabse chhota prime number?tap 👆
2 (aklauta even prime bhi)
Sabse chhota composite number?tap 👆
4
1 se 100 tak kitne primes?tap 👆
25 (1 se 50 tak: 15)
Kya 1 prime hai?tap 👆
Nahi — na prime, na composite
Kya 0 natural number hai?tap 👆
Nahi — whole hai, natural nahi
Co-prime numbers kya hote hain?tap 👆
Jinka HCF = 1, jaise (8, 15)
9 ki divisibility ka rule?tap 👆
Digits ka sum 9 se divide ho
11 ki divisibility ka rule?tap 👆
Odd − Even places ka sum = 0 ya 11 ka multiple
BODMAS ka full form?tap 👆
Bracket Of Division Multiplication Addition Subtraction
65² = ? (trick se)tap 👆
6×7 = 42 → 4225
7 ka unit-digit cycle?tap 👆
7, 9, 3, 1 (length 4)
Kya 91 prime hai?tap 👆
Nahi! 91 = 7 × 13 (famous trap)
36 × 25 = ? (trick se)tap 👆
3600 ÷ 4 = 900
| Fact / Question | Answer |
|---|---|
| Sabse chhota prime | 2 |
| Aklauta even prime | 2 |
| Sabse chhota composite | 4 |
| Sabse chhota odd prime | 3 |
| 1 se 50 tak primes | 15 |
| 1 se 100 tak primes | 25 |
| 1 se 100 tak ka sum | 5050 |
| BODMAS mein sabse pehle | Bracket (bar/line sabse pehle) |
| 'Of' ka matlab | Multiply (÷ se pehle hota hai) |
| ÷ aur × mein pehle kaun | Jo left mein pehle aaye |
| 0, 1, 5, 6 ka unit digit (kisi bhi power par) | Wahi digit |
| Pehle n odd numbers ka sum | n² |
Practice (exam-style)
Q1. Simplify: 12 + 8 ÷ 4 × 3 − 5 = ?
💡 8÷4=2 pehle, phir 2×3=6, phir 12+6−5 = 13. BODMAS!
Practice (exam-style)
Q2. The smallest prime number is:
💡 2 sabse chhota prime hai — 1 na prime hai na composite.
Practice (exam-style)
Q3. The unit digit of 745 is:
💡 7 ka cycle 7,9,3,1 — 45÷4 remainder 1 → cycle ka 1st digit = 7.
Practice (exam-style)
Q4. Which of the following numbers is divisible by 11?
💡 245157: odd places 7+1+4=12, even places 5+5+2=12, difference 0 → divisible.
Practice (exam-style)
Q5. The sum of the first 50 natural numbers is:
💡 n(n+1)/2 = 50×51/2 = 1275.
Practice (exam-style)
Q6. The place value of 5 in the number 25,634 is:
💡 5 thousands wali jagah par hai → 5 × 1000 = 5000 (face value hoti 5).
Practice (exam-style)
Q7. 999 × 47 = ?
💡 Trick: 47×1000 − 47 = 47000 − 47 = 46953.
Practice (exam-style)
Q8. Which of the following numbers is divisible by 9?
💡 4+5+7+8+3 = 27, jo 9 se divide hota hai → 45783 divisible.
Practice (exam-style)
Q9. Simplify: 25 − [8 − {6 − (4 − 2)}] = ?
💡 Andar se bahar: (4−2)=2 → 6−2=4 → 8−4=4 → 25−4 = 21.
Practice (exam-style)
Q10. How many prime numbers are there between 1 and 50?
💡 1–50 mein 15 primes hain (2 se 47 tak); 25 to 1–100 ka count hai.
Practice (exam-style)
Q11. 35² = ?
💡 Trick: 3×4 = 12, aage 25 → 1225.
Practice (exam-style)
Q12. The sum of the first 20 odd natural numbers is:
💡 Pehle n odd ka sum = n² = 20² = 400.
Practice (exam-style)
Q13. If the number 34x6 is divisible by 3, the smallest value of x is:
💡 3+4+x+6 = 13+x; 15 banane ke liye x = 2 (sum 3 ka multiple chahiye).
Practice (exam-style)
Q14. Simplify: 18 ÷ 3 of 2 = ?
💡 'Of' pehle: 3 of 2 = 6, phir 18÷6 = 3. (18÷3×2 hota to 12 aata!)
Numbers ke types yaad rakho: 1 na prime na composite, 2 aklauta even prime, 0 natural nahi. 1–100 mein 25 primes hain. Divisibility rules mein 3/9 = digit sum, 11 = odd−even places ka difference. Simplification ka boss hai BODMAS — bracket pehle, 'of' ÷ se pehle, aur ÷/× mein jo left mein pehle aaye. Sum formulas ratt lo: 1 se n tak = n(n+1)/2, pehle n odd = n². Tricks (n5² wala, ×999 wala) daily 5 sawalon par practice karo — speed apne aap banegi! 🚀