Viral Maths — Ch01: Important Products

📌 Why This Chapter Matters

Yeh chapter poori Viral Maths ki foundation hai — yahan diye gaye products, patterns, tables, aur squares agar cold yaad hain to baaki saare chapters (Addition se Fractions tak) bahut fast ho jaate hain.
Kuch bhi calculate karne se pehle brain ko "known constants" ka bank chahiye — yeh chapter wahi bank banata hai.
Daily revision zaroori hai: Doubles, Tables (2–25), Squares (1–100) — inhe test lene se pehle 5 min bhi revise karo.

⚡ QUICK RECALL

Is chapter mein 5 sections hain: Chaurahas (equal-product sets), Magic Number Patterns, Double of Numbers (1–100), Multiplication Tables (2–25), Squares (1–100).

🗂️ Chapter Index

Tab	Topic	What's Inside
2	Chaurahas	Different factor-pairs giving the same product
3	Magic Patterns	37-series, 16-series, 19-series, palindromes, repunits, 3-series & 9-series squares
4	Doubles 1–100	Full lookup table of 2n for n=1 to 100
5	Tables 2–25	Complete multiplication tables from 2× to 25×
6	Squares 1–100	Full squares list + digit-cross trick method
7	Master Table	Everything from this chapter in one place

🚦 Chaurahas — Equal-Product Memory Sets

⚡ QUICK RECALL — Concept

"Chauraha" = crossroads. Different multiplication pairs that land on the SAME product. Memorising these lets you instantly swap one pair for another mid-calculation — very useful for simplifying big multiplications and divisions.

Equal Product	Factor Pairs (the "crossroads")
108	12×9 = 27×4 = 36×3
180	45×4 = 20×9 = 36×5
144	24×6 = 36×4 = 18×8 = 16×9
216	24×9 = 36×6 = 12×18
192	24×8 = 96×2 = 12×16

Cube & Square Anchor Values

Cube of 6

6³ = 216 (matches the 216 chauraha above — 24×9 = 36×6 = 12×18 = 6³)

Square Anchors

12² = 144 | 25² = 625 | 12×25 = 300 | 24×25 = 600

⚠ EXAM TRAP

Chaurahas sirf "equal product" dikhate hain — yeh mat socho ki factor pairs khud equal hain. 12×9 aur 27×4 ke factors alag hain, sirf result (108) same hai.

Consecutive Number Products

Product	Result
12 × 13	156
13 × 14	182
14 × 15	210
15 × 16	240
16 × 17	272
17 × 18	306
18 × 19	342
12 × 25	300
24 × 25	600

37-Series (Multiples of 3)

37 ×	Result	37 ×	Result
3	111	18	666
6	222	21	777
9	333	24	888
12	444	27	999
15	555

⚡ QUICK RECALL

37 × (3k) always gives a 3-digit repdigit (k, k, k). Divide the multiplier by 3, repeat that digit thrice.

16-Series & 19-Series

16-series (×4 with trailing 6s)

16 × 4 = 64 166 × 4 = 664 1666 × 4 = 6664 16666 × 4 = 66664

19-series (×5 with trailing 9s)

19 × 5 = 95 199 × 5 = 995 1999 × 5 = 9995 19999 × 5 = 99995

Repunits & Palindromic Squares

Repunit squares (all 1's)

11 × 11 = 121 111 × 111 = 12321 1111 × 1111 = 1234321 11111 × 11111 = 123454321

Palindromic-zero squares

101 × 101 = 10201 1001 × 1001 = 1002001 10001 × 10001 = 100020001 100001 × 100001 = 10000200001 10101 × 10101 = 102030201 1001001 × 1001001 = 1002003002001

⚠ EXAM TRAP

Pattern sirf tab tak hold karta hai jab tak digit-carry na ho — 111111×111111 se pattern break ho sakta hai kyunki carry lag jaata hai. Exam mein 5-6 repunits tak hi trust karo.

3-Series & 9-Series Squares

3-series

33² = 1089 333² = 110889 3333² = 11108889

9-series

99² = 9801 999² = 998001 9999² = 99980001

⚡ QUICK RECALL — Pattern Logic

n nines' square = (n−1) nines | 8 | (n−1) zeros | 1. E.g. 9999² → 3 nines, 8, 3 zeros, 1 → 99980001.

👯 Double of Numbers — 1 to 100

⚡ QUICK RECALL

Yeh table roz revise karo — Addition, Ratio, aur Multiplication chapters mein "double karo" step baar-baar aata hai. Zabaani fluent hona chahiye.

n	2n	n	2n	n	2n	n	2n
1	2	26	52	51	102	76	152
2	4	27	54	52	104	77	154
3	6	28	56	53	106	78	156
4	8	29	58	54	108	79	158
5	10	30	60	55	110	80	160
6	12	31	62	56	112	81	162
7	14	32	64	57	114	82	164
8	16	33	66	58	116	83	166
9	18	34	68	59	118	84	168
10	20	35	70	60	120	85	170
11	22	36	72	61	122	86	172
12	24	37	74	62	124	87	174
13	26	38	76	63	126	88	176
14	28	39	78	64	128	89	178
15	30	40	80	65	130	90	180
16	32	41	82	66	132	91	182
17	34	42	84	67	134	92	184
18	36	43	86	68	136	93	186
19	38	44	88	69	138	94	188
20	40	45	90	70	140	95	190
21	42	46	92	71	142	96	192
22	44	47	94	72	144	97	194
23	46	48	96	73	146	98	196
24	48	49	98	74	148	99	198
25	50	50	100	75	150	100	200

📊 Multiplication Tables — 2 to 25

⚡ QUICK RECALL

Book ki requirement: tables 2 se 25 tak zabaani yaad hone chahiye, bina soche turant bol sako. Yeh Multiplication aur Division dono chapters ki backbone hai.

Table of 2

2×1	2
2×2	4
2×3	6
2×4	8
2×5	10
2×6	12
2×7	14
2×8	16
2×9	18
2×10	20

Table of 3

3×1	3
3×2	6
3×3	9
3×4	12
3×5	15
3×6	18
3×7	21
3×8	24
3×9	27
3×10	30

Table of 4

4×1	4
4×2	8
4×3	12
4×4	16
4×5	20
4×6	24
4×7	28
4×8	32
4×9	36
4×10	40

Table of 5

5×1	5
5×2	10
5×3	15
5×4	20
5×5	25
5×6	30
5×7	35
5×8	40
5×9	45
5×10	50

Table of 6

6×1	6
6×2	12
6×3	18
6×4	24
6×5	30
6×6	36
6×7	42
6×8	48
6×9	54
6×10	60

Table of 7

7×1	7
7×2	14
7×3	21
7×4	28
7×5	35
7×6	42
7×7	49
7×8	56
7×9	63
7×10	70

Table of 8

8×1	8
8×2	16
8×3	24
8×4	32
8×5	40
8×6	48
8×7	56
8×8	64
8×9	72
8×10	80

Table of 9

9×1	9
9×2	18
9×3	27
9×4	36
9×5	45
9×6	54
9×7	63
9×8	72
9×9	81
9×10	90

Table of 10

10×1	10
10×2	20
10×3	30
10×4	40
10×5	50
10×6	60
10×7	70
10×8	80
10×9	90
10×10	100

Table of 11

11×1	11
11×2	22
11×3	33
11×4	44
11×5	55
11×6	66
11×7	77
11×8	88
11×9	99
11×10	110

Table of 12

12×1	12
12×2	24
12×3	36
12×4	48
12×5	60
12×6	72
12×7	84
12×8	96
12×9	108
12×10	120

Table of 13

13×1	13
13×2	26
13×3	39
13×4	52
13×5	65
13×6	78
13×7	91
13×8	104
13×9	117
13×10	130

Table of 14

14×1	14
14×2	28
14×3	42
14×4	56
14×5	70
14×6	84
14×7	98
14×8	112
14×9	126
14×10	140

Table of 15

15×1	15
15×2	30
15×3	45
15×4	60
15×5	75
15×6	90
15×7	105
15×8	120
15×9	135
15×10	150

Table of 16

16×1	16
16×2	32
16×3	48
16×4	64
16×5	80
16×6	96
16×7	112
16×8	128
16×9	144
16×10	160

Table of 17

17×1	17
17×2	34
17×3	51
17×4	68
17×5	85
17×6	102
17×7	119
17×8	136
17×9	153
17×10	170

Table of 18

18×1	18
18×2	36
18×3	54
18×4	72
18×5	90
18×6	108
18×7	126
18×8	144
18×9	162
18×10	180

Table of 19

19×1	19
19×2	38
19×3	57
19×4	76
19×5	95
19×6	114
19×7	133
19×8	152
19×9	171
19×10	190

Table of 20

20×1	20
20×2	40
20×3	60
20×4	80
20×5	100
20×6	120
20×7	140
20×8	160
20×9	180
20×10	200

Table of 21

21×1	21
21×2	42
21×3	63
21×4	84
21×5	105
21×6	126
21×7	147
21×8	168
21×9	189
21×10	210

Table of 22

22×1	22
22×2	44
22×3	66
22×4	88
22×5	110
22×6	132
22×7	154
22×8	176
22×9	198
22×10	220

Table of 23

23×1	23
23×2	46
23×3	69
23×4	92
23×5	115
23×6	138
23×7	161
23×8	184
23×9	207
23×10	230

Table of 24

24×1	24
24×2	48
24×3	72
24×4	96
24×5	120
24×6	144
24×7	168
24×8	192
24×9	216
24×10	240

Table of 25

25×1	25
25×2	50
25×3	75
25×4	100
25×5	125
25×6	150
25×7	175
25×8	200
25×9	225
25×10	250

🔢 Squares — 1 to 100 (Full List)

n	n²	n	n²	n	n²	n	n²	n	n²
1	1	21	441	41	1681	61	3721	81	6561
2	4	22	484	42	1764	62	3844	82	6724
3	9	23	529	43	1849	63	3969	83	6889
4	16	24	576	44	1936	64	4096	84	7056
5	25	25	625	45	2025	65	4225	85	7225
6	36	26	676	46	2116	66	4356	86	7396
7	49	27	729	47	2209	67	4489	87	7569
8	64	28	784	48	2304	68	4624	88	7744
9	81	29	841	49	2401	69	4761	89	7921
10	100	30	900	50	2500	70	4900	90	8100
11	121	31	961	51	2601	71	5041	91	8281
12	144	32	1024	52	2704	72	5184	92	8464
13	169	33	1089	53	2809	73	5329	93	8649
14	196	34	1156	54	2916	74	5476	94	8836
15	225	35	1225	55	3025	75	5625	95	9025
16	256	36	1296	56	3136	76	5776	96	9216
17	289	37	1369	57	3249	77	5929	97	9409
18	324	38	1444	58	3364	78	6084	98	9604
19	361	39	1521	59	3481	79	6241	99	9801
20	400	40	1600	60	3600	80	6400	100	10000

Digit-Cross Trick — Find Any 2-digit Square Fast

Example: 68²

Step 1: Square of first digit: 6² = 36 Step 2: Square of second digit: 8² = 64 Step 3: Cross-multiply digits ×2: 6×8×2 = 96 Step 4: Write as 36 | 64, then add 96 (shifted one place) into the middle Step 5: 36|64 → add 9 to 36 (carry from 96), add 6 to 64's tens: (36+9)|(6+6→carry 1)|4 Final: 4624

⚠ EXAM TRAP

Cross-term (2×a×b) do-digit ho sakta hai — carry LEFT mein daalna mat bhoolo. Yeh common calculation mistake hai.

⚡ QUICK RECALL

Yeh wahi (a+b)² = a²+b²+2ab identity hai, jahan a=tens digit×10, b=units digit — Advance Maths ke Algebra chapter se connected hai.

📋 Master Table — Chapter 1 Summary

Section	Key Content
Chaurahas	108 (12×9=27×4=36×3), 144 (24×6=36×4=18×8=16×9), 180, 192, 216
Consecutive products	12×13=156 ... 18×19=342 \| 12×25=300 \| 24×25=600
37-series	37×3k = repdigit of (k,k,k)
16-series	16(6)×4 = 64(6...4) pattern
19-series	19(9)×5 = 95(9...5) pattern
Repunits	111...1 × 111...1 = palindrome (123...321 style)
Palindromic zeros	10...01 × 10...01 = 10...0201...0001 style
3-series squares	33²=1089, 333²=110889, 3333²=11108889
9-series squares	99²=9801, 999²=998001, 9999²=99980001
Doubles 1–100	Full lookup — 2n for every n
Tables 2–25	Complete ×1 to ×10 for every base 2–25
Squares 1–100	Full lookup + digit-cross trick for fast calculation

n	2n	n	2n	n	2n	n	2n
1	2	26	52	51	102	76	152
2	4	27	54	52	104	77	154
3	6	28	56	53	106	78	156
4	8	29	58	54	108	79	158
5	10	30	60	55	110	80	160
6	12	31	62	56	112	81	162
7	14	32	64	57	114	82	164
8	16	33	66	58	116	83	166
9	18	34	68	59	118	84	168
10	20	35	70	60	120	85	170
11	22	36	72	61	122	86	172
12	24	37	74	62	124	87	174
13	26	38	76	63	126	88	176
14	28	39	78	64	128	89	178
15	30	40	80	65	130	90	180
16	32	41	82	66	132	91	182
17	34	42	84	67	134	92	184
18	36	43	86	68	136	93	186
19	38	44	88	69	138	94	188
20	40	45	90	70	140	95	190
21	42	46	92	71	142	96	192
22	44	47	94	72	144	97	194
23	46	48	96	73	146	98	196
24	48	49	98	74	148	99	198
25	50	50	100	75	150	100	200

n	2n	n	2n	n	2n	n	2n
1	2	26	52	51	102	76	152
2	4	27	54	52	104	77	154
3	6	28	56	53	106	78	156
4	8	29	58	54	108	79	158
5	10	30	60	55	110	80	160
6	12	31	62	56	112	81	162
7	14	32	64	57	114	82	164
8	16	33	66	58	116	83	166
9	18	34	68	59	118	84	168
10	20	35	70	60	120	85	170
11	22	36	72	61	122	86	172
12	24	37	74	62	124	87	174
13	26	38	76	63	126	88	176
14	28	39	78	64	128	89	178
15	30	40	80	65	130	90	180
16	32	41	82	66	132	91	182
17	34	42	84	67	134	92	184
18	36	43	86	68	136	93	186
19	38	44	88	69	138	94	188
20	40	45	90	70	140	95	190
21	42	46	92	71	142	96	192
22	44	47	94	72	144	97	194
23	46	48	96	73	146	98	196
24	48	49	98	74	148	99	198
25	50	50	100	75	150	100	200

n	2n	n	2n	n	2n	n	2n
1	2	26	52	51	102	76	152
2	4	27	54	52	104	77	154
3	6	28	56	53	106	78	156
4	8	29	58	54	108	79	158
5	10	30	60	55	110	80	160
6	12	31	62	56	112	81	162
7	14	32	64	57	114	82	164
8	16	33	66	58	116	83	166
9	18	34	68	59	118	84	168
10	20	35	70	60	120	85	170
11	22	36	72	61	122	86	172
12	24	37	74	62	124	87	174
13	26	38	76	63	126	88	176
14	28	39	78	64	128	89	178
15	30	40	80	65	130	90	180
16	32	41	82	66	132	91	182
17	34	42	84	67	134	92	184
18	36	43	86	68	136	93	186
19	38	44	88	69	138	94	188
20	40	45	90	70	140	95	190
21	42	46	92	71	142	96	192
22	44	47	94	72	144	97	194
23	46	48	96	73	146	98	196
24	48	49	98	74	148	99	198
25	50	50	100	75	150	100	200