Vedic Math Guru

Gunakasamuchyah

गुणकसमुच्चयः

Description

Meaning: "The Samuccaya of the factors is equal to the Samuccaya of the product". This is very closely related to Gunitasamuchyah. While Gunitasamuchyah often refers specifically to the sum of coefficients (by setting variable=1), Gunakasamuchyah can be interpreted more broadly, sometimes referring to the sum of the digits of the numbers (often using the "casting out nines" or "digit sum" method) as a check for numerical multiplication.

Primary Application: Checking Numerical Multiplication using Digit Sums

This is a classic and simple method to quickly check if a multiplication of numbers is likely correct.

How it Works (Digit Sum Method / Casting Out Nines):

  1. Find the Digit Sum of the Multiplicand: Add all the digits of the first number. If the sum is a multi-digit number, add its digits again until you get a single digit (this is equivalent to finding the remainder when divided by 9, where a result of 9 is treated as 0 or 9 itself). Let this be DS1.
  2. Find the Digit Sum of the Multiplier: Do the same for the second number. Let this be DS2.
  3. Multiply the Digit Sums: Multiply DS1 x DS2. Find the digit sum of this product. Let this be DSP.
  4. Find the Digit Sum of the Product: Take the result of your original multiplication and find its digit sum. Let this be DSR.
  5. Verification: If DSP = DSR, then the multiplication is likely correct. If they are not equal, the multiplication is definitely incorrect.

"Casting Out Nines" aspect: When summing digits, any 9s or combinations of digits that sum to 9 can be ignored (cast out) as they don't affect the final single-digit sum (modulo 9).

Example: Check 123 x 45 = 5535.

  • Digit Sum of Multiplicand (123): 1+2+3 = 6. (DS1 = 6).
  • Digit Sum of Multiplier (45): 4+5 = 9. (DS2 = 9 or 0. Let's use 9 for now).
  • Multiply Digit Sums: DS1 x DS2 = 6 x 9 = 54. Digit sum of 54 is 5+4 = 9. (DSP = 9).
  • Digit Sum of Product (5535): 5+5+3+5 = 18. Digit sum of 18 is 1+8 = 9. (DSR = 9).

Since DSP (9) = DSR (9), the multiplication is likely correct.

Limitation: This method can only prove a multiplication incorrect; it cannot definitively prove it correct. For example, if digits are transposed in the answer (e.g., 5355 instead of 5535), the digit sum might still match, but the answer is wrong. However, it's a very good quick check.

Example 1: Check 12 x 13 = 156 using Gunakasamuchyah (digit sums).

Let's check if the multiplication 12 × 13 = 156 is correct using "Gunakasamuchyah", which in this context refers to checking with digit sums (also known as the "casting out nines" method).

Multiplication to check: 12 × 13 = 156

Step 1: Digit Sum of the First Number (Multiplicand)

Number: 12

Digit Sum (DS1) = 1 + 2 = 3

Step 2: Digit Sum of the Second Number (Multiplier)

Number: 13

Digit Sum (DS2) = 1 + 3 = 4

Step 3: Multiply the Digit Sums & Find its Digit Sum

DS1 × DS2 = 3 × 4 = 12

Digit sum of this product (12): 1 + 2 = 3.

Let this be DSP = 3.

Step 4: Digit Sum of the Proposed Answer

Proposed Answer: 156

Digit Sum (DSR) = 1 + 5 + 6 = 12

Digit sum of 12: 1 + 2 = 3.

Step 5: Verification

Digit Sum of (DS1 × DS2) => DSP = 3

Digit Sum of Proposed Answer => DSR = 3

Since DSP = DSR (3 = 3), the multiplication is likely correct!

Gunakasamuchyah (Digit Sum check) confirms our calculation! ✅

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