Vedic Maths cheat sheets

वैदिक गणित चीट शीट

Last-minute revision tables. Learn squares to 30, cubes to 15 and the fraction table first; they are what every other shortcut leans on.

The 16 sutras and where they help Which technique for which pattern Squares 1 to 60 Cubes 1 to 25 Fraction, decimal and percentage, 1/2 to 1/20 Divisibility rules Last digits and digit roots of squares and cubes Recurring cycles worth knowing

The 16 sutras and where they help · 16 सूत्र और उनका उपयोग

Sutra names are in standard transliteration. The right-hand column gives a typical use. सूत्रों के नाम मानक लिप्यंतरण में हैं। दाएँ स्तंभ में प्रत्येक का एक सामान्य उपयोग दिया गया है।

No. / क्र.	Sutra / सूत्र	Meaning / अर्थ	Used for / उपयोग
1	Ekādhikena Pūrveṇa	By one more than the previous one	Squares ending in 5, recurring decimals, osculation
2	Nikhilaṁ Navataścaramaṁ Daśataḥ	All from 9 and the last from 10	Subtraction from a base, near-base multiplication
3	Ūrdhva-Tiryagbhyām	Vertically and crosswise	Cross multiplication, × 11, factoring
4	Parāvartya Yojayet	Transpose and apply	Equations, division
5	Śūnyaṁ Sāmyasamuccaye	If the sum is the same, it is zero	Special equations
6	(Ānurūpye) Śūnyamanyat	If one is in ratio, the other is zero	Simultaneous equations
7	Saṅkalana-Vyavakalanābhyām	By addition and subtraction	Swapped-coefficient equations
8	Pūraṇāpūraṇābhyām	By completion or non-completion	Rounding and balancing
9	Calana-Kalanābhyām	Sequential motion (calculus-like)	Roots and polynomial derivatives
10	Yāvadūnam	Whatever the deficiency	Squares and cubes near a base
11	Vyaṣṭisamaṣṭiḥ	Part and whole	Special equations and factoring
12	Śeṣāṇyaṅkena Caramena	The remainders by the last digit	Recurring decimals
13	Sopāntyadvayamantyam	The ultimate and twice the penultimate	Special fractions in equations
14	Ekanyūnena Pūrveṇa	By one less than the previous one	Multiplying by 9, 99, 999
15	Gaṇitasamuccayaḥ	The product of the sums is the sum of the products	Digit-sum checks
16	Guṇakasamuccayaḥ	The factors of the sum are the sum of the factors	Factoring and verification

Which technique for which pattern · कौन-से पैटर्न के लिए कौन-सी तकनीक

You see / आप देखते हैं	Use / उपयोग करें	Example / उदाहरण
A number ending in 5, squared	n(n + 1) then 25	65² → 6 × 7 \| 25 = 4225
Equal tens parts, units add to 10	n(n + 1) \| product of units	43 × 47 → 20 \| 21 = 2021
× 11	Add neighbouring digits	76 × 11 = 836
Both numbers near 100 or 1000	Nikhilam: cross-add \| product of distances	97 × 94 = 9118
Square near a base	Number ± d \| d²	103² = 106 \| 09
× 9, 99, 999, 98	N × 10^k − N	47 × 99 = 4653
× 5, 25, 125	Halve, ÷ 4, ÷ 8, then shift	72 × 125 = 9000
A power of 10 minus N	All from 9, last from 10	1000 − 357 = 643
Any 2-digit × 2-digit	Cross multiplication	47 × 36 = 1692
Any 2- or 3-digit square	Duplex	63² = 3969
A perfect square under the root	Last digit + first pair	√5329 = 73
A perfect cube under the root	Last digit + leading part	∛226981 = 61
Check an answer	Digit sum (mod 9)	47 × 36: 2 × 0 = 0
x + 1/x given	p² − 2, p³ − 3p	p = 5 → 23, 110
Two equations, two unknowns	Crosswise formula	3x + 2y = 19, 5x + 4y = 33

Squares 1 to 60 · 1 से 60 तक के वर्ग

n / n	n² / n²	n / n	n² / n²
1	1	31	961
2	4	32	1024
3	9	33	1089
4	16	34	1156
5	25	35	1225
6	36	36	1296
7	49	37	1369
8	64	38	1444
9	81	39	1521
10	100	40	1600
11	121	41	1681
12	144	42	1764
13	169	43	1849
14	196	44	1936
15	225	45	2025
16	256	46	2116
17	289	47	2209
18	324	48	2304
19	361	49	2401
20	400	50	2500
21	441	51	2601
22	484	52	2704
23	529	53	2809
24	576	54	2916
25	625	55	3025
26	676	56	3136
27	729	57	3249
28	784	58	3364
29	841	59	3481
30	900	60	3600

Cubes 1 to 25 · 1 से 25 तक के घन

n / n	n³ / n³	n / n	n³ / n³
1	1	14	2744
2	8	15	3375
3	27	16	4096
4	64	17	4913
5	125	18	5832
6	216	19	6859
7	343	20	8000
8	512	21	9261
9	729	22	10648
10	1000	23	12167
11	1331	24	13824
12	1728	25	15625
13	2197

Fraction, decimal and percentage, 1/2 to 1/20 · भिन्न, दशमलव और प्रतिशत, 1/2 से 1/20

A bar over digits means they repeat. The 1/7 family (142857) and 1/13 family (076923) are worth learning as cycles. अंकों के ऊपर रेखा का अर्थ है कि वे दोहराए जाते हैं। 1/7 परिवार (142857) और 1/13 परिवार (076923) को चक्र के रूप में याद करना उपयोगी है।

Fraction / भिन्न	Decimal / दशमलव	Percentage / प्रतिशत
1/2	0.5	50%
1/3	0.3̄	33.33%
1/4	0.25	25%
1/5	0.2	20%
1/6	0.16̄	16.67%
1/7	0.1̄4̄2̄8̄5̄7̄	14.29%
1/8	0.125	12.5%
1/9	0.1̄	11.11%
1/10	0.1	10%
1/11	0.0̄9̄	9.09%
1/12	0.083̄	8.33%
1/13	0.0̄7̄6̄9̄2̄3̄	7.69%
1/14	0.07142857…	7.14%
1/15	0.06̄	6.67%
1/16	0.0625	6.25%
1/17	0.0588235294117647…	5.88%
1/18	0.05̄	5.56%
1/19	0.052631578947368421…	5.26%
1/20	0.05	5%

Divisibility rules · विभाज्यता नियम

By / से	Rule / नियम	Example / उदाहरण
2	Last digit is even	3,846
3	Digit sum divisible by 3	2,451: 2 + 4 + 5 + 1 = 12
4	Last two digits divisible by 4	7,316: 16
5	Last digit 0 or 5	1,245
6	Divisible by 2 and 3	4,128
7	1001 trick or osculator 5	91,728: 728 − 91 = 637 = 7 × 91
8	Last three digits divisible by 8	3,24,576: 576
9	Digit sum divisible by 9	7,218: 18
10	Last digit 0	3,450
11	Odd-place sum minus even-place sum is 0 or a multiple of 11	918082: 25 − 3 = 22
12	Divisible by 3 and 4	5,148
13	1001 trick or osculator 4	91,728: 637 = 13 × 49
25	Last two digits 00, 25, 50, 75	3,675
125	Last three digits divisible by 125	4,375

Last digits and digit roots of squares and cubes · वर्ग और घन के अंतिम अंक तथा अंक-मूल

Root ends in / मूल का अंतिम अंक	Square ends in / वर्ग का अंतिम अंक	Cube ends in / घन का अंतिम अंक
0	0	0
1	1	1
2	4	8
3	9	7
4	6	4
5	5	5
6	6	6
7	9	3
8	4	2
9	1	9

Recurring cycles worth knowing · जानने योग्य आवर्ती चक्र

Fraction / भिन्न	Decimal / दशमलव
1/7	0.142857 142857…
2/7	0.285714 285714…
3/7	0.428571 428571…
4/7	0.571428 571428…
5/7	0.714285 714285…
6/7	0.857142 857142…
1/11	0.0909…
1/13	0.076923 076923…
2/13	0.153846 153846…
1/19	0.052631578947368421 052631578947368421…

Learn the techniques on the Vedic Maths notes page, then drill them in practice.