Number System questions test how numbers behave under division, factorization, powers, and remainders. A clear grasp of number types, divisibility rules, factors, multiples, and cyclic patterns makes these questions much faster to solve.

Types of numbers

• Natural numbers: 1, 2, 3, and so on.
• Whole numbers: 0, 1, 2, 3, and so on.
• Integers: negative numbers, zero, and positive numbers.
• Rational numbers: numbers that can be written as p/q, where q is not zero.
• Irrational numbers: non-terminating, non-repeating decimals such as square root of 2.
• Real numbers: all rational and irrational numbers.

Prime numbers and factorization

A prime number has exactly two positive factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite.

• Express a composite number as a product of prime factors before solving factor, HCF, or LCM questions.
• For example, 360 = 2^3 x 3^2 x 5.
• The number of factors of p^a x q^b is (a + 1)(b + 1).

Factors, multiples, HCF, and LCM

• Factors divide a number exactly; multiples are produced by multiplying a number by integers.
• HCF is the greatest number that divides each given number.
• LCM is the smallest positive number divisible by every given number.
• For two positive numbers, HCF x LCM = product of the two numbers.
• Find HCF with the lowest powers and LCM with the highest powers in prime factorization.

Divisibility rules

• By 2: the last digit is even. By 5: the last digit is 0 or 5. By 10: the last digit is 0.
• By 3 or 9: the sum of digits is divisible by 3 or 9.
• By 4: the last two digits are divisible by 4. By 8: the last three digits are divisible by 8.
• By 11: the difference between the sums of alternating digits is 0 or a multiple of 11.

Remainders and modular arithmetic

When a number N is divided by d, N = dq + r, where 0 <= r < d. Work with remainders directly when dealing with large powers or sums.

• To add or multiply large numbers modulo d, reduce each number to its remainder first.
• For powers, look for the repeating cycle of remainders or unit digits.
• The unit digit cycles for powers of 2 are 2, 4, 8, 6 and repeat every 4 powers.

Decimals and fractions

• A fraction in simplest form has a terminating decimal only when its denominator has prime factors 2 and/or 5.
• Use place value to convert terminating decimals into fractions.
• For recurring decimals, use multiplication by powers of 10 and subtraction to form an equation.

How to solve questions

1. Identify whether the question concerns factors, divisibility, remainders, or digit patterns.
2. Use prime factorization for HCF, LCM, and factor-count questions.
3. Apply divisibility rules before performing long division.
4. Reduce large values to remainders and find cycles for powers.
5. Check that every remainder is non-negative and smaller than the divisor.