Number System — Extra Practice

The worth of each digit depends on where it sits — the basis of the whole number system.

We use only ten symbols — 0 to 9 — called digits. By placing them in different positions we build every number, small or big. The nine digits other than 0 are called significant digits.

Position matters: in 52 the 5 means fifty, but in 25 the same 5 means just five. Each place is worth ten times the one to its right: ones (1), tens (10), hundreds (100), thousands (1,000), ten thousands (10,000), lakhs (1,00,000), ten lakhs, crores, ten crores.

The table above takes apart the number 6,07,25,938. In the Indian system we put commas after the first three digits from the right and then after every two digits (thousand, lakh, crore).

The families of numbers — which sits inside which — and the special roles of 0, 1 and 2.

The best way to remember the names is to picture boxes inside boxes: natural numbers sit inside whole numbers, whole numbers inside integers, integers inside rational numbers, and rational numbers inside real numbers.

• 0 is not a natural number but is a whole number. It is neither positive nor negative.
• 1 is neither prime nor composite. The smallest prime is 2, and it is the only even prime.
• Co-prime numbers need not be prime: (8, 15) are co-prime though both are composite.
• $\frac{22}{7}$ is only an approximation of $\pi$; $\pi$ itself is irrational.

Each divisor has its own quick test — look at the last digits, or add the digits.

These tests work because some of 10, 100, 1000… divide exactly by the divisor. 100 is divisible by 4, so for 4 only the last two digits matter.

Combined tests: a number divisible by two co-prime numbers is divisible by their product. So 6 = 2×3, 12 = 3×4, 15 = 3×5, 18 = 2×9, 24 = 3×8, 36 = 4×9. Careful: you cannot test 12 with 2 and 6 (they are not co-prime) — use 3 and 4.

Test for 7: double the last digit and subtract it from the rest of the number; if the result is 0 or a multiple of 7 the number is divisible by 7 (repeat if needed). Test for 13: add four times the last digit to the rest. A neat fact: $7\times 11\times 13=1001$, so every six-digit number of the form $abcabc$ (like 482,482) is divisible by all three.

The “least number to add or subtract” and “greatest or smallest n-digit number” questions.

Dividing a number by a divisor leaves a remainder. That remainder tells you how far the number is from the multiple just below it and the multiple just above it. To reach the multiple below, subtract the remainder; to reach the one above, add (divisor − remainder).

Greatest and smallest n-digit numbers

The greatest n-digit number is 99…9 (for 4 digits, 9,999) and the smallest is 100…0 (1,000). Use the subtract rule on the greatest; the add rule on the smallest.

Finding remainders without dividing, and how remainders behave when you add or multiply.

Every division can be written as dividend = divisor × quotient + remainder, and the remainder is always between 0 and divisor−1. The divisibility tests double as remainder tests: digit sum for 9, last two digits for 4, last three for 8, alternating sum for 11.

When multiplying, the units digit depends only on the last digits.

When two numbers are multiplied, the units digit of the product depends only on the units digits of the two numbers — the other digits go into tens and above. So take only the last digit of each number, multiply, and keep only the last digit at every step.

The units digit of $a^n$ runs in a short cycle — know the cycle length and any power becomes easy.

Keep raising a number to higher powers and the units digit changes, then starts repeating after a few steps. Example: $7^1=7$, $7^2=49$ (units 9), $7^3=343$ (units 3), $7^4=2401$ (units 1), and $7^5$ ends in 7 again. So 7 has a cycle of four: 7, 9, 3, 1.

Method: divide the power by the cycle length (1, 2 or 4). If the remainder is 1, 2 or 3, take that position in the cycle; if the remainder is 0, take the last digit of the cycle. Since 1, 2 and 4 all divide 4, dividing the power by 4 works in every case.

Decide whether a number is prime by testing only a few small primes.

A prime has exactly two factors: 1 and itself. There are 25 primes up to 100:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

The shortcut: to test whether $N$ is prime, divide by every prime up to $\sqrt{N}$. If none divides it, $N$ is prime. Why? If $N$ has a factor, its partner factor is at most $\sqrt{N}$.

Break a number into primes and count its factors, add them, or count special kinds of factors.

Every number can be written as a product of primes in exactly one way, e.g. $360=2^3\times 3^2\times 5$. That form tells us everything about its factors: each factor picks a power of 2 from 0 to 3 (4 choices), a power of 3 from 0 to 2 (3 choices) and a power of 5 from 0 to 1 (2 choices).

• Even factors need the power of 2 to be at least 1: $a\times (b+1)(c+1)$. Odd factors have power 0: $(b+1)(c+1)$.
• A number with an odd count of factors is a perfect square (all its prime powers are even).
• Ways to write $N$ as a product of two factors = (total factors) ÷ 2 (when $N$ is not a perfect square).

Formulas for 1 to n, even and odd numbers, their squares and cubes — and how to apply them.

The story of Gauss is famous: to add 1 to 100 he paired the first and last numbers (1+100, 2+99, …). Each pair sums to 101 and there are 50 pairs: $50\times 101=5050$. The same idea gives the formula $\frac{n(n+1)}{2}$.

Questions that give the sum of consecutive numbers and ask for the smallest or largest, plus a few useful properties.

When numbers are evenly spaced (consecutive numbers, consecutive even numbers) their average is the middle number. So: middle = sum ÷ count, and the others lie at equal steps on both sides.

A few useful properties

• The product of three consecutive numbers is always divisible by 6; of four consecutive, by 24.
• $n^3-n=(n-1)n(n+1)$ is always divisible by 6.
• A perfect square ends only in 0, 1, 4, 5, 6 or 9 — never in 2, 3, 7 or 8.
• If $p$ divides two numbers it also divides their sum and difference.

The table takes you straight from the wording to the method.

Marks in number system are lost to carelessness, not to difficulty.

One-rule questions.

Each question needs one rule. Aim for 30–40 seconds.

Two-step, exam-level questions.

Questions at the hard end of competitive exams.