The 13 Sub-Sutras
Upa-Sutras are auxiliary aphorisms — corollaries of the main sutras for specific situations.
आनुरूप्येण
Ānurūpyeṇa
"Proportionately"
Scale problems up or down to a simpler base, solve, then scale back.
48×47 (work near base 50)Related: nikhilam-navatascaramam-dasatah, ekadhikena-purvena
शिष्यते शेषसंज्ञः
Śiṣyate Śeṣasaṃjñaḥ
"The remainder remains constant"
Remainders in division follow predictable patterns that allow rapid completion.
100 ÷ 7 — track the remainderRelated: paraavartya-yojayet
आद्यमाद्येन
Ādyamādyena
"The first by the first and the last by the last"
In polynomial products, the leading terms and constant terms multiply simply — spot them first.
Factor 6x²+13x+6 using first×first and last×lastRelated: urdhva-tiryagbyham
केवलैः सप्तकं गुण्यात्
Kevalaiḥ Saptakaṃ Guṇyāt
"For 7, multiply by 143"
1/7 = 142857/999999. Multiplying by 7's magic number 143 relates to the 1/7 decimal cycle.
1/7 recurring decimal cycleRelated: ekadhikena-purvena
वेष्टनम्
Veṣṭanaṃ
"By osculation"
Test divisibility using the osculator — multiply the last digit by the osculator and add to the remaining number, repeat until single digit.
Is 91 divisible by 7?Related: shesanyankena-charamena
यावदूनं तावदूनीकृत्य
Yāvadūnaṃ Tāvadūnīkṛtya
"Whatever the deficiency, lessen it still further by that much"
Subtract the deficiency twice when cubing near a base — extends Yaavadunam to cubes.
98³ using extended YaavadunamRelated: yaavadunam
अन्त्ययोर्दशकेऽपि
Antyayordaśakepi
"When the last digits sum to 10"
If two numbers have the same prefix and their last digits sum to 10, the multiplication is nearly as fast as squaring ending in 5.
83 × 87 (8×9=72, 3×7=21)74 × 76Related: ekadhikena-purvena
अन्त्ययोरेव
Antyayoreva
"Only the last terms"
Certain products only need the last digits computed — the rest follow from the leading terms.
12×14 — last digits determine units digitRelated: urdhva-tiryagbyham
समुच्चयगुणितः
Samuccayaguṇitaḥ
"The sum multiplied"
The sum of terms can be factored out — reduces multi-term multiplication to factor × sum.
Factor 3x²+15x+18Related: gunitasamuchyah
लोपन-स्थापनाभ्याम्
Lopanasthāpanābhyāṃ
"By alternate elimination and retention"
Solve equations by temporarily dropping one variable, solving for the other, then restoring it.
3-variable system by elimination-retentionRelated: sankalana-vyavakalanabhyam
विलोकनम्
Vilokanam
"By mere observation"
Many problems solve themselves by simple inspection — train the eye to see structure directly.
Spot 4×9=36 by observation aloneगुणितसमुच्चयः समुच्चयगुणितः
Guṇitasamuccayaḥ Samuccayaguṇitaḥ
"The product of the sum equals the sum of the products"
The digit-root verification rule: digitRoot(A×B) = digitRoot(A) × digitRoot(B). Extends Gunitasamuchyah.
Verify 12×13=156 via digit sums: 3×4=12→3; 156→3 ✓Related: gunitasamuchyah
ध्वजाङ्क
Dhvajāṅka
"On the flag"
The flag digit in Paraavartya division — the sub-digit that gets transposed and applied at each step.
1234÷12 with flag digit 2Related: paraavartya-yojayet