Single-Digit Modulo 9
The Digital Root ($\text{DR}$) of a number is the single-digit sum ($1 \text{ to } 9$) obtained by repeatedly adding all of its digits until one digit remains.
Casting Out 9s Model
Since $10 \equiv 1 \pmod 9$, any $9$ or digit pair adding up to $9$ contributes $0$ to the digital sum:
- • $\text{DR}(79452) \implies$ Strike $9$, strike $(7+2=9)$ & $(4+5=9) \implies \mathbf{9}$
- • $\text{DR}(48319) \implies$ Strike $9$, strike $(8+1=9) \implies 4+3 = \mathbf{7}$
- • Addition & multiplication digital roots match the result's digital root!
In multiple-choice competitive exams (CAT, MAH MBA CET, SNAP, NMAT), you do not need to calculate exact answers if you can eliminate the 3 wrong options:
Addition Law
Example: $\text{DR}(457 + 389) = \text{DR}(7 + 2) = 9$.
Multiplication Law
Example: $\text{DR}(63 \times 42) = \text{DR}(9 \times 6) = 9$.
Negative DR Trick
Example: $\text{DR}(32 - 78) = 5 - 6 = -1 + 9 = 8$.
4-Step Option Elimination Framework
Step 1
Unit Digit ($\text{Mod } 10$)Step 2
Digital Root ($\text{Mod } 9$)Step 3
Boundary Range BoundingStep 4
Divisibility Check ($\text{Mod } 4, 11$)OPTION ELIMINATION SHORTCUT RULES
Digit Striking Shortcut
Saves precious seconds by skipping repetitive additions when calculating the digital sum.
Upper & Lower Range Bounding
Prevents selecting options that are numerically out of possible range magnitude.
SOLVED EXAMPLES (LEVEL 0 TO HARD)
Q: Find the product of $346 \times 729$ from options: (A) 252,234 (B) 252,238 (C) 251,824 (D) 252,734.
Step 1 (Unit Digit): $6 \times 9 = 54 \implies$ Ends in 4. Option B (ends in 8) is ELIMINATED!
Step 2 (Digital Root): $\text{DR}(729) = 9 \implies \text{DR}(\text{Product}) = 9$.
Step 3 (Option DRs): A = $9$, C = $4$, D = $5$. Only Option A has $\text{DR} = 9$!
Q: Evaluate $84321 - 39485$.
Step 1: $\text{DR}(84321) = 8+4+3+2+1 = 18 \implies 9$.
Step 2: $\text{DR}(39485) = 3+9+4+8+5 = 29 \implies 11 \implies 2$.
Step 3: $\text{DR}(\text{Diff}) = 9 - 2 = 7$. Correct option must have $\text{DR} = 7$ (e.g., $44836$).
Q: Find the value of $(123 \times 456) + 789$.
Step 1: $\text{DR}(123) = 6$, $\text{DR}(456) = 6$, $\text{DR}(789) = 6$.
Step 2: $\text{DR}(6 \times 6 + 6) = \text{DR}(36 + 6) = \text{DR}(42) = 6$.
Step 3: Actual value = $56088 + 789 = 56877$. $\text{DR}(56877) = 5+6+8+7+7 = 33 \implies \mathbf{6}$. Verified!
Mistake 1
Direct Division of Digital Roots
Never divide digital roots directly ($A / B$). Rearrange into multiplication $A = B \times C$ before checking DR!
Mistake 2
Relying Solely on Digital Root
If two options have the same DR, always combine DR with the Unit Digit filter to break the tie.
Mistake 3
Ignoring Negative Subtraction
Forgetting to add $+9$ when $\text{DR}(A) - \text{DR}(B)$ turns negative.
1. Find the Digital Root of 987654321.
Answer: All digits form pairs adding to 9 $(1+8, 2+7, 3+6, 4+5)$ and one 9 $\implies \mathbf{9}$.
2. What is the digital root of $53^2$?
Answer: $\text{DR}(53) = 8 \implies \text{DR}(8^2) = \text{DR}(64) = 10 \implies \mathbf{1}$. ($53^2 = 2809 \implies \text{DR} = 1$).
3. Verify if $78 \times 96 = 7488$ using digital root.
Answer: $\text{DR}(78) = 6, \text{DR}(96) = 6 \implies \text{DR}(36) = 9$. $\text{DR}(7488) = 27 \implies \mathbf{9}$. Verified!
FREQUENTLY ASKED QUESTIONS
❓ What is the Digital Root of a number and how does Casting Out 9s work?
Digital Root is the single-digit sum obtained by repeatedly adding digits until 1-9 remains. Casting Out 9s allows you to ignore any 9s or digit combinations summing to 9.
❓ Can Digital Root be used for division questions?
Not directly in division form. Rearrange A / B = C into A = B * C, then verify DR(A) = DR(B * C).
❓ What is the 4-step option elimination matrix for MBA exams?
1) Unit Digit filtering (mod 10). 2) Digital Root verification (mod 9). 3) Boundary Range Bounding. 4) Divisibility test (mod 4, 7, 11).