Number System Questions for SSC CGL

Classification of Numbers

Every number falls into categories. Natural numbers start from 1. Whole numbers include 0. Integers include negatives. Rational numbers can be written as fractions. Prime numbers have exactly 2 factors (1 and itself). Composite numbers have more than 2 factors. 1 is neither prime nor composite. In SSC CGL, you'll often need to quickly identify whether a number is prime or composite, especially in divisibility and factor-based questions.

• Prime: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 (memorize up to 50)
• 2 is the only even prime number
• Composite: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30...
• Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256...
• Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000...
• Co-prime numbers: Two numbers with HCF = 1 (e.g., 7 and 9)

Divisibility Rules (Speed Hack)

Divisibility rules let you instantly check if a number divides another without long division. This is critical in SSC CGL because questions often ask 'which number divides X?' or 'find remainder when Y is divided by Z'. Knowing these rules saves 30-60 seconds per question. Memorize all 12 rules below—they appear in nearly every paper.

• Divisible by 2: Last digit is even (0, 2, 4, 6, 8)
• Divisible by 3: Sum of digits is divisible by 3
• Divisible by 4: Last two digits form a number divisible by 4
• Divisible by 5: Last digit is 0 or 5
• Divisible by 6: Divisible by both 2 and 3
• Divisible by 7: No simple rule—use division or modular arithmetic
• Divisible by 8: Last three digits form a number divisible by 8
• Divisible by 9: Sum of digits is divisible by 9
• Divisible by 10: Last digit is 0
• Divisible by 11: Alternating sum of digits is divisible by 11 (e.g., 121: 1-2+1=0, divisible by 11)
• Divisible by 12: Divisible by both 3 and 4
• Divisible by 25: Last two digits are 00, 25, 50, or 75

HCF (GCD) and LCM

HCF is the highest common factor—the largest number that divides two or more numbers. LCM is the least common multiple—the smallest number divisible by two or more numbers. SSC CGL frequently asks: 'Find HCF of 48 and 64' or 'Find LCM of 12, 15, 20'. The prime factorization method is fastest. For HCF, take the lowest power of common prime factors. For LCM, take the highest power of all prime factors.

• Prime factorization method works for all problems
• HCF × LCM = Product of the two numbers (only for two numbers)
• HCF is always less than or equal to the smaller number
• LCM is always greater than or equal to the larger number
• If two numbers are co-prime, HCF = 1 and LCM = product of numbers
• For three or more numbers, use prime factorization; the shortcut formula doesn't apply

Digit-Based Problems

SSC CGL loves questions like 'A two-digit number is 4 times the sum of its digits' or 'Find a number such that when digits are reversed, the new number is 27 more'. These require setting up equations. If the number is AB (where A is tens digit, B is units digit), the value is 10A + B. Reverse is 10B + A. Use this framework to solve all digit problems.

• Two-digit number = 10a + b (where a is tens digit, b is units digit)
• Three-digit number = 100a + 10b + c
• Reversed two-digit number = 10b + a
• Sum of digits of 10a + b is (a + b)
• Difference when reversed: (10b + a) - (10a + b) = 9(b - a)
• Always check that digits are between 0-9 and leading digit is not 0

Remainder and Modular Arithmetic

When a number is divided by another, we get quotient and remainder. SSC CGL asks: 'What is the remainder when 2^100 is divided by 7?' or 'Find the remainder when 12345 is divided by 11'. For large powers, use Fermat's Little Theorem or cyclicity of remainders. For divisibility by 11, use the alternating sum rule. Modular arithmetic notation: a ≡ b (mod n) means a and b have the same remainder when divided by n.

• Remainder is always less than the divisor
• Remainder when a + b is divided by n = (remainder of a + remainder of b) mod n
• Remainder when a × b is divided by n = (remainder of a × remainder of b) mod n
• Remainder when a^n is divided by m often follows a cycle—find the pattern
• For 2^n mod 5: cycle is 2, 4, 3, 1, 2, 4, 3, 1... (period 4)
• For 3^n mod 10: cycle is 3, 9, 7, 1, 3, 9, 7, 1... (period 4)

• Confusing HCF and LCM: HCF is smaller, LCM is larger. Many aspirants reverse these in a panic.
• Forgetting that 1 is neither prime nor composite. This trips up classification questions.
• Misapplying the divisibility rule for 11: The alternating sum must start from the right (units place), not the left.
• In digit problems, forgetting to check that digits are valid (0-9) and leading digit is not 0. A solution like a=12, b=5 is invalid.
• Using HCF × LCM = a × b for three or more numbers. This formula ONLY works for exactly two numbers.
• Not recognizing cyclicity in remainder problems and attempting brute-force calculation for 2^100 mod 7, wasting time.

• PRIME UP TO 50: 'Two, Three, Five, Seven, Eleven, Thirteen, Seventeen, Nineteen, Twenty-Three, Twenty-Nine, Thirty-One, Thirty-Seven, Forty-One, Forty-Three, Forty-Seven.' Sing it to a tune.
• HCF vs LCM: HCF = 'High Common Factor' (but actually LOWER number). LCM = 'Least Common Multiple' (but actually HIGHER number). Think: HCF is the 'common ground' (smaller), LCM is the 'common ceiling' (larger).
• Divisibility by 11: 'Odd minus Even' — subtract digits in even positions from digits in odd positions (right to left).
• Digit reversal difference: Always 9 times the difference of digits. (10b + a) - (10a + b) = 9(b - a). Memorize this shortcut.
• Modular arithmetic: 'Remainder rules are like algebra—you can add and multiply remainders separately, then take remainder again.'

• SSC CGL Quant typically has 1-2 straightforward divisibility/HCF-LCM questions (Tier 1, easy) and 1-2 digit-based or remainder problems (Tier 2, medium). Solve the easy ones first in under 1 minute each.
• Divisibility rules are your speed weapon. A question like 'Which of these divides 54321?' can be answered in 10 seconds if you know the rule for 11, but takes 2 minutes with long division.
• Recent SSC CGL papers (2022-2024) show a trend toward remainder and cyclicity problems (e.g., 'Find last digit of 7^2024'). Practice modular arithmetic and cyclicity patterns.
• Digit-based problems often appear in Tier 2. They require careful algebra setup. Spend 2-3 minutes on these; don't guess. The constraint 'digits are in ratio 1:2:3' or 'sum of digits is 18' usually gives a unique answer.
• In the exam, if a divisibility rule doesn't come to mind, skip and return. Don't waste time on division. The rule is always faster than calculation.