Verification & Sanity Check
Divisibility (3, 9, 11) & Multiplication Sanity Check
To test divisibility or find remainders quickly, sum the digits (for 3 and 9) or alternate digit signs (for 11). To check large 3-digit multiplication A × B = C, multiply the remainders of A and B: they MUST equal the remainder of C! If remainders differ, the answer is guaranteed wrong in 5 seconds.
Mathematical Symmetry & Identity
$$\text{rem}_M(A) \times \text{rem}_M(B) \equiv \text{rem}_M(A \times B) \pmod{M}$$
Visual Mental Pipeline
Step-by-Step Execution Rules
- 1 Divisibility by 3 & 9 (Digital Sum): Add the digits. If digit sum is divisible by 3 (or 9), the number is too. The remainder of the digit sum equals the remainder of the full number! Example: 5,423 → 5+4+2+3 = 14 → 1+4 = 5. Remainder mod 9 is 5; remainder mod 3 is 5 ÷ 3 = rem 2.
- 2 Divisibility by 11 (Alternating Sum): From right to left, alternate signs: (+ units - tens + hundreds - thousands...). If result is negative, add 11. Example: 8,426 → +6 -2 +4 -8 = 0 → Divisible by 11! Example 2: 7,394 → +4 -9 +3 -7 = -9 → -9 + 11 = 2 (Remainder 2).
- 3 Checking 3-Digit Multiplication (Casting Out 9s): If A × B = C, then (rem(A) × rem(B)) mod 9 MUST EQUAL rem(C) mod 9. Both sides must have the exact same remainder! Example: 483 × 729 = 352,107 → rem(483)=6, rem(729)=0 → 6 × 0 = 0. Product 352,107 has rem 0. 0 = 0, verified!
Worked Examples & Thought Process
Find remainder: 7,456 ÷ 9 and 7,456 ÷ 3
Example 1
Step 1: Sum digits: 7 + 4 + 5 + 6 = 22
Step 2: Sum again: 2 + 2 = 4 → Remainder mod 9 is 4
Step 3: For mod 3: 4 ÷ 3 = Remainder 1
= Mod 9: Rem 4 | Mod 3: Rem 1
Find remainder: 8,392 ÷ 11
Example 2
Step 1: Right to left: (+2) - (9) + (3) - (8)
Step 2: Alternating sum: 2 - 9 + 3 - 8 = -12
Step 3: -12 mod 11: -12 + 22 = 10 (or -1 + 11 = 10)
= Remainder 10
Check: 342 × 615 = 210,330 (Correct or Incorrect?)
Example 3
Step 1: LHS: rem(342) = 3+4+2=9→0; rem(615) = 6+1+5=12→3
Step 2: Multiply remainders: 0 × 3 = 0 (Expected mod 9)
Step 3: RHS: 2+1+0+3+3+0 = 9 → 0. 0 = 0 (Match!)
= CORRECT (Passed Mod 9 check)
Check: 624 × 318 = 198,332 (Correct or Incorrect?)
Example 4
Step 1: LHS: rem(624) = 6+2+4=12→3; rem(318) = 3+1+8=12→3
Step 2: Multiply remainders: 3 × 3 = 9 → 0 (Expected mod 9)
Step 3: RHS: 1+9+8+3+3+2 = 26 → 8 ≠ 0 (MISMATCH!)
= INCORRECT (Caught in 5s without multiplying)
Interactive Checkpoint 1
Try It: Rapid Remainder Finder (Mod 3, 9, 11)
Use the digit sum or alternating sum trick to find the remainder of this number:
MODULO 9 CHECK
54,238
What is the remainder when divided by 9?
Interactive Checkpoint 2
Try It: 3-Digit × 3-Digit Multiplication Verifier
Do not compute the full multiplication! Calculate the digital root (mod 9) of both factors, multiply them, and check if they match the product's digital root:
Is This Product Correct or Incorrect?
483 × 729 = 352,107
Use casting out 9s: does \(\text{rem}(A) \times \text{rem}(B) \equiv \text{rem}(C) \pmod 9\)?
Ready to Hone Your Speed?
Put this shortcut into practice with 10 procedural, timed questions. Sharpen your reaction time and track your accuracy.