गुणितसमुच्चयः
Guṇitasamuccayaḥ
"The product of the sum is the sum of the products"
What this sutra solves
Sanity-check any multiplication answer in a couple of seconds.
You worked out 97 × 94 = 9118 on paper and want to be sure before relying on it.
Digit roots: 7 × 4 → 1, and 9118 → 1. They match, so the answer passes the check.
Use digit roots to check whether a multiplication result is plausible.
Verify a multiplication answer
Use digit roots to check whether a multiplication result is plausible.
⚡ Speed Advantage
2× faster with Vedic Mathematics
Best for
- • Verifying multiplication, division, and squaring results
Use when
- • After any multiplication or squaring to sanity-check the answer
Avoid when
- • As the primary computation method
Intuition
The digit sum of a product equals the digit sum of the product of the individual digit sums — a verification tool.
Story Mode
The Digital Fingerprint
Every number has a fingerprint: its digit root. When you multiply two numbers, their fingerprints multiply too. If the answer's fingerprint doesn't match, there's an error. This ancient check is the world's fastest arithmetic proof-reader.
Vedic vs conventional
No conventional equivalent — this is purely a verification technique.
Applications
Digit sum verification of multiplication
Verify any multiplication result using digit sums in 3 mental steps.
Common Mistakes to Avoid
Digit sum 9 and 0 are equivalent (mod 9 = 0)
Wrong approach
Correct approach
Why this happens
💡 Students forget the modular nature of digit-root checking.
Why It Works
A number and its digit sum have the same remainder mod 9:
Multiplication preserves remainders:
Therefore digit roots must match:
If the digit-root check fails, the answer is definitely wrong. If it passes, the answer is plausible but still not fully proven.
Digit root is a homomorphism modulo 9: ). This is because the digit root tracks the number modulo 9.