1. Introduction

Divisibility is the single most exam-productive sub-topic inside Number System, and Number System itself is one of the highest-weightage areas in SSC CGL, CHSL, MTS, and Railway RRB NTPC/Group D quantitative sections. In a typical SSC CGL Tier-I or Tier-II paper, you can expect 2 to 4 questions built directly on divisibility rules, missing-digit problems, factorial zero-counting, or remainder shortcuts — and indirectly, divisibility reasoning is the backbone of at least another 3–5 questions on HCF-LCM, simplification, and number-series problems. RRB NTPC and Group D papers show a very similar pattern, often repeating the same question formats year after year (missing digit for divisibility by 11, trailing zeros in a factorial, remainder using the “sum of digits” trick).

The difficulty level ranges from straightforward (apply the rule for 3, 9, or 11 to a given number) in CHSL/MTS/Group D to moderately tricky in CGL Tier-II and NTPC, where you must combine two rules (e.g., divisibility by 8 and 9 to find two missing digits), work with algebraic divisibility of expressions like a^n - b^n, or compute the highest power of a prime dividing a large factorial. None of this requires deep theory — it requires memorized rules, a few standard formulas (especially Legendre’s formula for factorials), and speed. That is exactly what this chapter builds.

Divisibility connects tightly to the rest of Number System: HCF/LCM problems frequently reduce to “is this divisible by that,” remainder-and-cyclicity problems use divisibility to simplify huge powers, and “forming numbers from digits” problems (a CGL Tier-II favourite) are almost always secretly divisibility problems in disguise. Master this chapter and several other chapters become noticeably easier.

How to use this chapter for maximum score: first internalize the rule table in Section 2 until you can apply any single rule (2 through 19, plus 25) in under 5 seconds by sight. Then move to the algebraic divisibility patterns (a^n± b^n) and the factorial-based formulas (trailing zeros, highest power of a prime) in Sections 3–5, since these two areas alone account for a disproportionate share of “tricky” divisibility questions on real papers. Sections 6–10 train you to combine rules and reason through mixed word problems — exactly the skill tested in Tier-II and NTPC-level papers. Finally, attempt Practice Set A cold (no notes) to confirm the basics are automatic, then attempt Set B under a strict time limit (aim for under 12 minutes for all 15 questions) to simulate real exam pressure. Review every solution in the Answer Key even for questions you got right — the worked method often reveals a faster route than the one you used.