Divisibility Rules 2 to 13: Quick Division Tricks for Aptitude

Aptitude division questions cost most students 40 to 60 seconds each; the divisibility shortcuts below cut that to under 10.

Where Division Slows You Down

Quantitative aptitude sections in campus placement tests (TCS NQT, AMCAT, eLitmus, and company-specific assessments) typically include 15 to 25 questions in 20 to 30 minutes. Number-properties and factor questions account for three to five of those. Every second spent doing long division is a second not available for the next question.

The solution is not to divide faster. It is to avoid dividing at all. A divisibility rule turns a multi-step calculation into a five-second mental check. The rules below cover divisors 2 through 13, with verified examples for each one.

Why 13? Divisors up to 13 cover the prime factorization of numbers below 200, which is the range most aptitude problems operate in. Knowing these rules means you can factor any candidate number mentally without reaching for a scratch pad.

Divisibility Rules at a Glance: 2 Through 10

Wikipedia’s divisibility rule reference lists rules for dozens of divisors. For campus aptitude tests, the divisors from 2 to 10 cover the majority of number-property and factor questions.

The table below applies each rule to one number.

Divisor	Rule	3,528 check	Result
2	Last digit even (0, 2, 4, 6, 8)	Last digit 8	Yes
3	Digit sum divisible by 3	3+5+2+8=18, 18÷3=6	Yes
4	Last two digits form a multiple of 4	28÷4=7	Yes
5	Last digit is 0 or 5	Last digit 8	No
6	Divisible by both 2 and 3	Even; digit sum 18	Yes
8	Last three digits form a multiple of 8	528÷8=66	Yes
9	Digit sum divisible by 9	18÷9=2	Yes
10	Last digit is 0	Last digit 8	No

All results verified: 3528÷2=1764, 3528÷3=1176, 3528÷4=882, 3528÷6=588, 3528÷8=441, 3528÷9=392.

Rule 6 combines two checks you can run in parallel: glance at the last digit for evenness (Rule 2), then add the digits for Rule 3. Both checks take under three seconds together.

The Rule 8 sub-check: if the hundreds digit of the last three digits is even, the last two digits must themselves be divisible by 8. If the hundreds digit is odd, add 4 to the last two digits and check the result for divisibility by 8. For 528, the hundreds digit is 5 (odd), so add 4 to 28 to get 32. 32÷8=4. Alternatively, simply divide 528 by 8 directly if the three-digit mental calculation is faster for you.

Divisibility by 7: Step by Step

The rule for 7 is the one most students avoid because it looks complicated. It is not.

The rule: take the last digit, double it, subtract from the remaining number. If the result is 0 or divisible by 7, the original number is too. Repeat the step if the result still has more than two digits.

Worked examples:

• 161: Last digit 1, double = 2. Remaining: 16. 16 minus 2 = 14. 14 = 7 times 2. Divisible. (Verification: 161 = 7 times 23.)
• 203: Last digit 3, double = 6. Remaining: 20. 20 minus 6 = 14. Divisible. (Verification: 203 = 7 times 29.)
• 346: Last digit 6, double = 12. Remaining: 34. 34 minus 12 = 22. 22 is not divisible by 7. (Verification: 346 divided by 7 is approximately 49.4.)
• 1,001: Last digit 1, double = 2. Remaining: 100. 100 minus 2 = 98. 98 = 7 times 14. Divisible. (Verification: 1,001 = 7 times 143, confirmed as 7 times 140 plus 7 times 3 = 980 plus 21 = 1,001.)

A common misprint in older aptitude study materials applies the subtraction in reverse: subtracting the remaining number from the doubled last digit instead of subtracting the doubled last digit from the rest. Always double the last digit, then subtract that doubled value from the remaining number, not the other way around.

The rule also works by repeated application. For a five-digit number, apply it twice. Each step reduces the digit count by one.

Divisibility by 11 and 13

Divisibility by 11

As described in Khan Academy’s divisibility tests review: starting from the rightmost digit, alternately subtract and add digits as you move left. If the total is 0 or a multiple of 11, the number is divisible by 11.

Worked examples:

• 121: Starting from right: 1 minus 2 plus 1 = 0. Divisible. (Verification: 121 = 11 times 11.)
• 2,728: Starting from right: 8 minus 2 plus 7 minus 2 = 11. Divisible. (Verification: 2,728 = 11 times 248, confirmed as 11 times 200 plus 11 times 48 = 2,200 plus 528 = 2,728.)
• 1,234: Starting from right: 4 minus 3 plus 2 minus 1 = 2. Not divisible by 11. (Verification: 1,234 divided by 11 is approximately 112.2.)

The sign-flip error: some references describe the rule starting from the leftmost digit and say “subtract even-position digits from odd-position digits.” That phrasing produces the same answer as long as the position numbering is applied consistently. The easiest version for most people is simply to alternate subtract-add starting from the rightmost digit, moving left.

Divisibility by 13

The rule: multiply the last digit by 4, add to the remaining number. Repeat if the result still has more than two digits.

Worked examples:

• 91: Last digit 1, times 4 = 4. Remaining: 9. 9 + 4 = 13. Divisible. (Verification: 91 = 7 times 13.)
• 299: Last digit 9, times 4 = 36. Remaining: 29. 29 + 36 = 65. 65 = 5 times 13. Divisible. (Verification: 299 = 13 times 23, confirmed as 13 times 20 plus 13 times 3 = 260 plus 39 = 299.)
• 143: Last digit 3, times 4 = 12. Remaining: 14. 14 + 12 = 26. 26 = 2 times 13. Divisible. (Verification: 143 = 11 times 13.)
• 350: Last digit 0, times 4 = 0. Remaining: 35. 35 + 0 = 35. 35 is not divisible by 13. (Verification: 350 divided by 13 is approximately 26.9.)

The common mistake with Rule 13 is using 9 as the multiplier instead of 4. Nine is the negative osculator, which works in a “multiply and subtract” variant and produces the same answer. The “multiply by 4 and add” version is simpler for timed use because the addition never creates a negative intermediate result.

Quick Fraction Approximation for Data Interpretation

Divisibility checks tell you whether a number divides exactly. Data Interpretation asks a different question: what is this fraction approximately equal to? A two-step proportional technique answers that without long division.

The method:

• Step 1: Round both the numerator and denominator to the nearest friendly values and identify the resulting simple ratio.
• Step 2: Adjust the denominator precisely to a round number, then scale the numerator proportionally.

Worked example re-derived from first principles:

• Target fraction: 482 divided by 855.
• Step 1: 482 rounds to 500; 855 rounds to 900. Quick ratio: 500 divided by 900 = 5 divided by 9, approximately 0.556.
• Step 2: Denominator changes from 855 to 900, an increase of 45. To hold the 5-to-9 ratio, numerator increases by 45 times (5 divided by 9) = 25. Adjusted fraction: (482 + 25) divided by (855 + 45) = 507 divided by 900.
• 507 divided by 900 = 0.5633. Actual value: 482 divided by 855 = 0.5637. Difference: 0.0004.

Any multiple-choice option set spaced 0.02 or more apart will return the same answer with this approximation. An error of 0.0004 does not change the chosen option.

The same proportional logic applies to Time and Work aptitude questions, where work rates are expressed as fractions and combined using addition-of-rates. Both techniques rely on the same core insight: express a ratio in its simplest form first, then adjust for precision only where needed.

For the campus placement evaluation test, Data Interpretation and number-properties questions appear across all major assessment sections. Having divisibility checks and fraction approximation as practiced reflexes, not just understood concepts, is the practical difference between a 15-second solver and a 60-second one.

The Bank of America aptitude test includes Data Interpretation with harder fraction comparisons than most service-company papers. The approximation method is especially useful there.

The approximation discipline in the fraction section above has a direct parallel in applied machine learning: pick a simpler model, measure how far off it is, and decide if the error is acceptable for the task. TinkerLLM lets you run that loop with real language models at ₹299. If the 0.0004 gap in the 482-divided-by-855 example made intuitive sense, the AI version of the same idea will too.