Number System (RRB)

The number system is the bedrock of every quantitative section — and in RRB exams it shows up as fast, formula-driven questions on LCM/HCF, divisibility, unit digits and trailing zeros. This lesson builds the intuition behind each tool so you can solve these in seconds, not minutes.

Definition: The number system classifies numbers (natural, whole, integer, rational, real) and studies their properties — divisibility, factors, primes, and the patterns hidden in powers and factorials.

Classifying numbers

Numbers nest inside one another like boxes:

• Natural numbers (N): 1, 2, 3, … (counting numbers).
• Whole numbers (W): 0, 1, 2, 3, … (naturals plus 0).
• Integers (Z): …, −2, −1, 0, 1, 2, … (whole numbers plus negatives).
• Rational numbers (Q): any number expressible as p/q with q ≠ 0 (includes fractions and terminating/recurring decimals).
• Real numbers (R): all rationals plus irrationals (like √2, π) — every point on the number line.

Why it matters: Many "is this prime/composite/rational?" questions test only whether you know the right box a number belongs to.

Divisibility and the LCM–HCF link

Divisibility rules (for 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) let you test factors without dividing — for example, a number is divisible by 3 if its digit-sum is divisible by 3, and by 11 if the alternating digit-sum is divisible by 11.

The single most-tested identity is:

LCM × HCF = product of the two numbers (i.e. LCM(a,b) × HCF(a,b) = a × b).

Two numbers are co-prime when their HCF = 1 (they share no common factor except 1, e.g. 8 and 15).

Worked example:
Question: The HCF of two numbers is 6 and their LCM is 36. If one number is 12, find the other.
Solution:
Step 1: Use LCM × HCF = product of the numbers → 36 × 6 = 12 × (other).
Step 2: 216 = 12 × (other), so other = 216 ÷ 12 = 18.
Conclusion: The other number is 18 (check: HCF(12,18)=6, LCM(12,18)=36 ✓).

Unit-digit (cyclicity) of large powers

When you raise a number to a high power, only the last digit follows a repeating cycle. Memorise the cycle lengths:

• Ending in 2: cycle of 4 → 2, 4, 8, 6.
• Ending in 3: cycle of 4 → 3, 9, 7, 1.
• Ending in 7: cycle of 4 → 7, 9, 3, 1.
• Ending in 8: cycle of 4 → 8, 4, 2, 6.
• Ending in 9: cycle of 2 → 9, 1.
• Ending in 0, 1, 5, 6: the unit digit stays the same for every power.
• Ending in 4: cycle of 2 → 4, 6.

The trick: for a 4-cycle, divide the exponent by 4 and use the remainder to pick the digit (remainder 0 means the last digit of the cycle).

Worked example:
Question: Find the unit digit of 7^123.
Solution:
Step 1: 7 has a 4-cycle: 7, 9, 3, 1.
Step 2: 123 ÷ 4 leaves remainder 3.
Step 3: The 3rd term in the cycle is 3.
Conclusion: The unit digit of 7^123 is 3.

Trailing zeros in a factorial

A trailing zero comes from a factor of 10 = 2 × 5. In any factorial there are far more 2s than 5s, so the number of trailing zeros equals the count of factors of 5 in n!:

Number of zeros = ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + …

Worked example:
Question: How many trailing zeros does 100! have?
Solution:
Step 1: ⌊100/5⌋ = 20.
Step 2: ⌊100/25⌋ = 4.
Step 3: ⌊100/125⌋ = 0; stop.
Step 4: Add: 20 + 4 = 24.
Conclusion: 100! ends in 24 zeros.

Common misconception: Students count factors of 10 directly, or count 2s. Always count 5s — they are the scarcer factor and therefore the limiting one.

The five RRB-favourite question types

1. Find the LCM or HCF of two (or more) numbers.
2. Find the smallest number to ADD to make a number divisible by X (remainder method).
3. Find the largest n-digit number divisible by Y (divide, subtract the remainder).
4. Find the unit digit of a large power (cyclicity).
5. Identify a number as prime or composite, or as co-prime.

:::compare HCF vs LCM

:::keypoints Key points

• Number boxes nest: N ⊂ W ⊂ Z ⊂ Q ⊂ R.
• LCM × HCF = product of the two numbers — the most-tested identity.
• Co-prime means HCF = 1.
• Unit digits repeat in cycles; 0,1,5,6 never change.
• For a 4-cycle, use exponent ÷ 4 remainder to pick the unit digit.
• Trailing zeros in n! = count of 5s = ⌊n/5⌋ + ⌊n/25⌋ + …
• Memorise the first 20 primes and the divisibility rules for instant filtering.
  :::
  :::memory
• "Trailing zeros are starved of fives" — count factors of 5, never 2.
  :::
  :::recap
• Classify a number first; many questions just test the right box.
• LCM × HCF = a × b cracks most LCM/HCF problems.
• Unit digit of a power depends only on its cycle and the exponent's remainder.
• Trailing zeros of a factorial = number of 5-factors.
• Speed comes from memorised primes and divisibility rules.
  :::