Zero-Remainder Division
A number $N$ is divisible by $d$ if dividing $N$ by $d$ leaves a remainder of zero ($N \equiv 0 \pmod d$).
Co-Prime Factor Rule
To test divisibility by a composite number $C = a \times b$, test divisibility by $a$ and $b$ where $\gcd(a, b) = 1$:
- • Divisibility by 6 $\implies$ Test 2 AND 3
- • Divisibility by 72 $\implies$ Test 8 AND 9 ($\gcd(8,9) = 1$)
- • Divisibility by 88 $\implies$ Test 8 AND 11 ($\gcd(8,11) = 1$)
Divisibility tests appear directly in unknown digit questions ($5x38y$ divisible by 72) and indirectly in remainder theorems, factorials, and option elimination:
Rules for 2, 4, 8, 16
• 2: Last 1 digit divisible by 2 ($0,2,4,6,8$)
• 4: Last 2 digits divisible by 4 ($2^2$)
• 8: Last 3 digits divisible by 8 ($2^3$)
Example: $54136 \implies 36 \div 4 = 9 \implies \text{Divisible by 4}$.
Rules for 3 & 9
• 3: Sum of all digits is divisible by 3
• 9: Sum of all digits is divisible by 9
Example: $48213 \implies \text{Sum} = 18 \implies \text{Divisible by 3 & 9}$.
Rules for 5, 25 & 10
• 5: Last digit is 0 or 5
• 25: Last 2 digits are 00, 25, 50, 75
• 10: Last digit is 0
Example: $78925 \implies \text{Ends in } 25 \implies \text{Divisible by 25}$.
Rule for 7 (Double & Subtract)
Example: $343 \implies 34 - 2(3) = 28 \div 7 = 4 \implies \text{Divisible!}$
Rule for 11 (Odd - Even)
Difference must be $0$ or multiple of $11$. Example: $1331 \implies (1+3) - (3+1) = 0$.
Rule for 13 (Multiply 4 & Add)
Example: $169 \implies 16 + 4(9) = 52 \div 13 = 4 \implies \text{Divisible!}$
Composite Divisibility Cheat Sheet
SHORTCUT RULES & UNKNOWN DIGIT STRATEGY
Solving Unknown Digits ($x, y$) for 72
2) Use Rule of 9 on digit sum to find x.
Always test Rule of 8 first because it only depends on the last 3 digits, giving a unique value for $y$!
Rule of 99 (Pair Sum Shortcut)
If the sum of 2-digit pairs is divisible by 99, the number is divisible by 99.
SOLVED EXAMPLES (LEVEL 0 TO HARD)
Q: Find the single-digit value of $x$ if $745x82$ is completely divisible by 9.
Step 1: Sum of digits = $7 + 4 + 5 + x + 8 + 2 = 26 + x$.
Step 2: For $26 + x$ to be a multiple of 9, the next multiple of 9 after 26 is $27$.
Step 3: $26 + x = 27 \implies x = 1$.
Q: If the 8-digit number $425x36y2$ is divisible by 72, find the value of $x + y$ (assuming $y$ is the largest possible digit).
Step 1 (Rule of 8): Last 3 digits $6y2 \div 8$. For $y=9 \implies 692 / 8$ (rem 4). For $y=7 \implies 672 / 8 = 84 \implies \text{Largest } y = 7$.
Step 2 (Rule of 9): Sum of digits = $4 + 2 + 5 + x + 3 + 6 + 7 + 2 = 29 + x$.
Step 3: Next multiple of 9 is $36 \implies 29 + x = 36 \implies x = 7$.
Step 4: $x + y = 7 + 7 = 14$.
Q: What is the remainder when the 100-digit number $123456789101112...99$ is divided by 9?
Step 1: Remainder when $N$ is divided by 9 equals remainder of the sum of digits of $N$ divided by 9.
Step 2: Digits from 1 to 99: Sum of units digits = $10 \times (1+2+...+9) = 10 \times 45 = 450$.
Step 3: Sum of tens digits = $10 \times (1+2+...+9) = 450$. Total sum = $900$.
Step 4: $900 \pmod 9 = 0 \implies \text{Remainder = 0}$.
Mistake 1
Using Non-Coprime Factors
Testing 12 using 2 and 6 instead of 3 and 4. A number divisible by 2 and 6 is not necessarily divisible by 12 (e.g. 18).
Mistake 2
Rule of 8 Digit Overflow
Testing only the last 2 digits for 8 instead of the last 3 digits.
Mistake 3
Sign Alternation Errors in 11
Miscounting odd vs even position places from left to right instead of right to left.
1. Is 35728 divisible by 4 and 8?
Answer: Last 2 digits = 28 (divisible by 4). Last 3 digits = 728 ÷ 8 = 91 (divisible by 8). Yes to both!
2. Find $k$ if $91876k2$ is divisible by 8.
Answer: $6k2 \div 8 \implies 632 / 8 = 79$ or $672 / 8 = 84 \implies k = 3$ or $7$.
3. If $8x5146y$ is divisible by 88, find $x \times y$.
Answer: $46y \div 8 \implies y = 4$. Odd-Even for 11: $(4+4+5+8) - (6+1+x) = 21 - (7+x) = 14 - x \implies x = 3$. Product $x \times y = 3 \times 4 = 12$.
FREQUENTLY ASKED QUESTIONS
❓ How do you test if a number is divisible by 7?
Double the last digit and subtract it from the remaining truncated number. If the resulting difference is divisible by 7 (or 0), the original number is divisible by 7.
❓ What is the rule for composite divisibility (e.g. divisibility by 72 or 88)?
Break the composite number into co-prime factors a and b (where gcd(a,b)=1). For 72, use 8 and 9 (since gcd(8,9)=1). A number is divisible by 72 if and only if it is divisible by both 8 and 9.
❓ What is the divisibility rule for 11?
Calculate the difference between the sum of digits at odd positions and the sum of digits at even positions. If the absolute difference is 0 or a multiple of 11, the number is divisible by 11.