Vedic Mathematics - 3 Digit Multiplication | Vedic Maths Multiplication Tricks
Vedic Mathematics provides incredibly fast mental calculation methods for multiplication, and one of the most popular and versatile sutras (aphorisms) for multiplying 3-digit numbers is Urdhva-Tiryagbhyam (उर्ध्व तिर्यग्भ्याम्), which translates to "Vertically and Crosswise".
The Pattern for 3-digit × 3-digit Multiplication
Write the two numbers one above the other:
texta b c
× d e f
────────────
You calculate 5 groups of digits from right to left (exactly like 2-digit had 3 groups):
Rightmost digit (units place): vertical → c × f
Second digit (tens place): crosswise → (b × f) + (c × e)
Middle digit (hundreds place): crosswise all pairs → (a × f) + (b × e) + (c × d)
Fourth digit (thousands place): crosswise → (a × e) + (b × d)
Leftmost digit (ten-thousands place): vertical → a × d
Carry over remainders (10 or more) to the next column on the left, just like normal addition.
Example 1: 234 × 312 (very commonly used example)
Write it like this:
text2 3 4
× 3 1 2
────────────
Now compute step by step from right to left:
Step 1 (units): 4 × 2 = 8
Answer so far: … 8
Step 2 (tens): (3 × 2) + (4 × 1) = 6 + 4 = 10
Write 0, carry 1
Answer so far: … 08 (carry 1 to next)
Step 3 (hundreds): (2 × 2) + (3 × 1) + (4 × 3) = 4 + 3 + 12 = 19 + carry 1 = 20
Write 0, carry 2
Answer so far: … 008 (carry 2)
Step 4 (thousands): (2 × 1) + (3 × 3) = 2 + 9 = 11 + carry 2 = 13
Write 3, carry 1
Answer so far: … 3008** (carry 1)
Step 5 (ten-thousands): 2 × 3 = 6 + carry 1 = 7
Final answer: 72,648 (7 3 0 0 8 → 73008, wait — actually 72648)
Correct result: 234 × 312 = 72,648
Example 2: 456 × 321
text4 5 6
× 3 2 1
────────────
Step 1: 6 × 1 = 6
Step 2: (5 × 1) + (6 × 2) = 5 + 12 = 17 → write 7, carry 1
Step 3: (4 × 1) + (5 × 2) + (6 × 3) = 4 + 10 + 18 = 32 + carry 1 = 33 → write 3, carry 3
Step 4: (4 × 2) + (5 × 3) = 8 + 15 = 23 + carry 3 = 26 → write 6, carry 2
Step 5: 4 × 3 = 12 + carry 2 = 14
Answer: 1 4 6 3 7 6 → 146,376
(You can verify: 456 × 300 = 136,800; 456 × 20 = 9,120; 456 × 1 = 456 → total 146,376 ✓)
Example 3: 746 × 323 (famous one-line style)
text7 4 6
× 3 2 3
────────────
Step 1: 6 × 3 = 18 → write 8, carry 1
Step 2: (4 × 3) + (6 × 2) = 12 + 12 = 24 + carry 1 = 25 → write 5, carry 2
Step 3: (7 × 3) + (4 × 2) + (6 × 3) = 21 + 8 + 18 = 47 + carry 2 = 49 → write 9, carry 4
Step 4: (7 × 2) + (4 × 3) = 14 + 12 = 26 + carry 4 = 30 → write 0, carry 3
Step 5: 7 × 3 = 21 + carry 3 = 24
Answer: 240,758 (2 4 0 7 5 8)
Many experienced Vedic math practitioners do this entire calculation mentally or in one flowing line after sufficient practice.
Quick Summary of the Steps (Right to Left)
Position → Calculation
1 (units) → c × f
2 (tens) → b×f + c×e
3 (hundreds) → a×f + b×e + c×d
4 (thousands) → a×e + b×d
5 (ten-thousands) → a × d
Add carry from previous step each time.
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