Aptitude · Lesson 8 of 9
Number series & number system
Spotting patterns in series, divisibility rules, HCF and LCM.
Series patterns to test, in order
- Differences (and differences of differences).
- Multiplication/division by a constant or changing factor (×2, ×3…).
- Squares/cubes (n², n³, n² ± 1) and primes.
- Alternating series — two interleaved patterns.
Number system essentials
- Divisibility: by 3 or 9 if the digit sum is; by 4 if the last two digits are; by 8 if the last three are; by 11 if the alternating digit sum difference is 0 or a multiple of 11.
HCF × LCM = product of the two numbers.- LCM is used for “when will events coincide” questions; HCF for “largest equal groups”.
Example: Find the next number: 3, 7, 15, 31, 63, ?
- Differences: 4, 8, 16, 32 (doubling).
- Next difference = 64 → 63 + 64. (Also: each term = previous × 2 + 1.)
Answer: 127
Example: Three buses leave together and return every 12, 18 and 30 minutes. After how many minutes do they next leave together?
- LCM(12, 18, 30): 12 = 2²·3, 18 = 2·3², 30 = 2·3·5.
- LCM = 2² · 3² · 5.
Answer: 180 minutes (3 hours)
Key points
- Check differences first, then ratios, then squares/cubes.
- HCF × LCM = product of two numbers.
- Digit-sum rule for 3 and 9.
Quick check
Next in the series 1, 4, 9, 16, 25, ?
They are squares: 1², 2², 3², 4², 5², so 6² = 36.
Practice task
Find the missing term: 2, 6, 12, 20, 30, ?
Show answer
Terms are n(n+1): 1×2, 2×3, 3×4, 4×5, 5×6 → next 6×7 = 42.
Interview corner
Is 1,23,456 divisible by 3?
Digit sum = 1+2+3+4+5+6 = 21, which is divisible by 3 — so yes.
HCF of two numbers is 6 and LCM is 72; one number is 18. Find the other.
6 × 72 / 18 = 24.
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