Number System: Concepts, Formulas, and Clear Examples

Number system MCQs cover divisibility, remainders, units digits, factors, and base conversions. Modular thinking beats long division.

Learn divisibility tests for 2,3,4,5,8,9,11 and $(a^n \bmod m)$ cycles for units digit.

For remainders when dividing by 9 or 3, digit sum shortcuts often decide the option.

Divisibility tests

• By 11: alternate digit sum difference divisible by 11.
• By 8: last three digits divisible by 8.

Remainders

Division algorithm

$$N = dq + r,\ 0 \le r < d$$

Mod addition

$$(a+b)\bmod m = [(a\bmod m)+(b\bmod m)]\bmod m$$

Units digit cycles

• Powers of 2 cycle 2,4,8,6; powers of 3 cycle 3,9,7,1 every 4 exponents mod 4.

Example — Units digit of 7^202

$202\bmod 4=2$. Cycle position 2 for 7 → units digit 9.

Factors and count

Number of factors

$$N=p_1^{a}p_2^{b}\cdots \Rightarrow (a+1)(b+1)\cdots$$

LCM/HCF tie-in

• Product of two numbers = HCF × LCM.
• Perfect squares have odd factor count.

Base conversion

• Binary to decimal: sum bit × $2^k$.
• Decimal to base: repeated divide by base, read remainders up.

Step-by-step method for exam questions

1. Classify: divisibility, remainder, units digit, or factors.
2. Apply test or mod cycle; reduce exponent mod cycle length.
3. For large N mod small d, simplify using digit sum or binomial.
4. Count factors from prime powers.
5. Eliminate options failing parity/divisibility.
6. Select answer.

Formula cheat sheet

Frequently asked questions

Negative remainders?

Adjust to $0\le r<m$ for final answer.

0 mod m?

Means divisible by m.

Last two digits?

Work mod 100; use mod 4 and mod 25 if needed.

Prime check fast?

Trial divide by primes up to $\sqrt{n}$.

Practice on this platform

Use 4 practice sets (10 MCQs, detailed solution after each answer) and 2 timed mocks (20 minutes) on this topic to convert formulas into speed and accuracy.