Number system: LCM, HCF, remainders and divisibility without the guesswork

Number-system questions reward a toolkit, not cleverness. Divisibility rules, the LCM–HCF identity, remainder shortcuts and unit-digit cycles cover the overwhelming majority of what SSC, railway and clerical papers ask. Learn the handful of tools below and learn which question calls for which, and the topic becomes some of the fastest marks on the paper.

Divisibility rules worth memorising

These convert a division into a glance:

• 2, 5, 10 — look at the last digit (even; 0 or 5; 0).
• 4 — last two digits form a multiple of 4. 8 — last three digits do.
• 3 and 9 — the digit sum is a multiple of 3, or of 9.
• 6 — divisible by both 2 and 3.
• 11 — the alternating sum of digits (from the right: minus, plus, minus) is 0 or a multiple of 11.

Most “which of these is divisible by…” and “find the missing digit so that…” questions are just one of these rules applied once.

HCF and LCM: the product rule

The identity that unlocks the topic: for two numbers,

first × second = LCM × HCF

So a question giving any three of those four is a one-line rearrangement. Two other templates recur:

• “Greatest number that divides a, b, c leaving the same remainder” — it is the HCF of the differences (a − b), (b − c).
• “Least number that when divided by a, b, c leaves remainder r each time” — it is LCM(a, b, c) + r. If the remainders differ but the gap to the divisor is constant, take LCM − (that constant gap).

Knowing which template you are in matters more than the arithmetic, which is small once the setup is right.

Remainders: digit sums and the product rule

Two ideas handle most remainder questions:

• Digit tests give the remainder, not just divisibility. The remainder on division by 9 is the remainder of the digit sum; by 11, the remainder of the alternating sum.
• The remainder of a product is the product of the remainders (then reduced again). To find (17 × 23) mod 5, take 17 mod 5 = 2 and 23 mod 5 = 3, multiply to get 6, and 6 mod 5 = 1. This keeps enormous products manageable, and it is the whole trick behind most “find the remainder when a big expression is divided by n” questions.

Unit digit of a power: the cycle of four

Every digit’s powers cycle with a period that divides 4:

• 2: 2, 4, 8, 6 — then repeats. 3: 3, 9, 7, 1. 7: 7, 9, 3, 1. 8: 8, 4, 2, 6.
• 4 and 9 have period 2; 0, 1, 5, 6 never change.

To find a unit digit, divide the exponent by 4 and use the remainder to pick the position in the cycle (a remainder of 0 means the last entry). For 2^34, since 34 leaves remainder 2 on division by 4, the unit digit is the second in 2’s cycle — 4.

Factors, quickly

Write the number as a product of primes, N = p^a × q^b × .... Then the number of factors is (a + 1)(b + 1)..., and the sum of factors multiplies the series for each prime. For 72 = 2^3 × 3^2, the factor count is (3 + 1)(2 + 1) = 12. Questions asking how many divisors a number has are testing exactly this, and nothing else.

Where the marks leak

Confusing the two “greatest/least number” templates. Greatest divisor uses HCF of differences; least dividend uses LCM plus remainder. Read which quantity is being asked for.

Reducing a remainder incompletely. After multiplying remainders, reduce again — 6 mod 5 is 1, not 6.

Miscounting the power cycle. A remainder of 0 on dividing the exponent by 4 points at the fourth entry, not the first.

Forgetting to reduce a ratio of quantities before applying a rule — small numbers, fewer slips.

Practice that transfers

Number-system speed depends on clean mental arithmetic, so keep the calculation-speed drill running while you practise, and use simplification and approximation to handle the messy expressions these questions hide inside. Then test the toolkit under time in a full-length mock test or the daily general awareness and quant quiz.