Number System Questions for AGNIVEER ARMY

Types of Numbers

Numbers are classified into several types. Natural numbers start from 1. Whole numbers start from 0. Integers include negatives too. Rational numbers can be written as p/q (like 3/4). Irrational numbers cannot (like root 2, pi). Prime numbers have exactly 2 factors: 1 and themselves. Composite numbers have more than 2 factors. Remember: 1 is neither prime nor composite.

• Natural numbers: 1, 2, 3, ... (no zero, no negatives)
• Whole numbers: 0, 1, 2, 3, ...
• Integers: ...-2, -1, 0, 1, 2, ...
• Prime numbers below 20: 2, 3, 5, 7, 11, 13, 17, 19 — exactly 8 primes
• 1 is neither prime nor composite — exam favourite trap
• 2 is the only even prime number

Divisibility Rules

Divisibility rules let you quickly check if a number divides another without long division. These are heavily tested in Agniveer. Learn rules for 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 by heart. Rule for 11 is most commonly asked: alternating digit sum difference must be 0 or divisible by 11.

• Div by 2: last digit even (0,2,4,6,8)
• Div by 3: sum of all digits divisible by 3
• Div by 4: last two digits divisible by 4
• Div by 5: last digit 0 or 5
• Div by 6: divisible by both 2 and 3
• Div by 8: last three digits divisible by 8
• Div by 9: sum of all digits divisible by 9
• Div by 11: (sum of odd-position digits) minus (sum of even-position digits) = 0 or multiple of 11

LCM and HCF

HCF (Highest Common Factor) is the largest number that divides two or more numbers exactly. LCM (Lowest Common Multiple) is the smallest number that is a multiple of two or more numbers. Key relation: LCM x HCF = Product of two numbers. This formula works only for two numbers, not three — a common exam trap. In word problems, use HCF when splitting/grouping and LCM when finding common time or meeting point.

• LCM x HCF = First number x Second number (only for 2 numbers)
• HCF always divides LCM completely
• HCF of fractions = HCF of numerators / LCM of denominators
• LCM of fractions = LCM of numerators / HCF of denominators
• If two numbers are co-prime, HCF = 1 and LCM = their product

Unit Digit (Last Digit) Tricks

Unit digit questions ask: what is the last digit of a large power like 7 to the power 85? Each digit follows a cycle when raised to powers. Memorize the cycle lengths. This is a pure pattern-memory question — once you know the cycles, it takes 20 seconds.

• Digits 0,1,5,6 always end in same digit for any power (cycle of 1)
• Digit 4: cycle is 4,6,4,6 (odd power = 4, even power = 6)
• Digit 9: cycle is 9,1,9,1 (odd power = 9, even power = 1)
• Digits 2,3,7,8: cycle of 4. Divide power by 4, use remainder
• Remainder 1 = 1st in cycle, Remainder 2 = 2nd, Remainder 3 = 3rd, Remainder 0 = 4th

Factors and Remainders

Number of factors of a number is found by prime factorization. Write the number as product of primes with powers, add 1 to each power, then multiply. Remainder problems often use direct division or the pattern/cycle method. Sum of factors and product of factors also appear occasionally.

• Prime factorize: N = a^p x b^q x c^r
• Total factors = (p+1)(q+1)(r+1)
• Sum of factors formula: [(a^(p+1)-1)/(a-1)] x [(b^(q+1)-1)/(b-1)] ...
• If remainder when N is divided by D is R, then N = DQ + R
• Remainder of a sum = sum of individual remainders (then reduce if needed)

• Applying LCM x HCF = product rule to 3 numbers — it only works for exactly 2 numbers
• Thinking 1 is a prime number — it is not; exam sets this trap often
• Forgetting that 2 is the only even prime — all other primes are odd
• In unit digit cycles, treating remainder 0 as the 0th term instead of the 4th (last) term in cycle
• Confusing HCF of fractions formula with LCM of fractions — numerator/denominator rule is opposite for both

• PRIME trick for divisibility by 11: Odd positions Minus Even positions = 0 or 11 multiple. Think ODD MINUS EVEN = ELEVEN.
• For unit digits, remember 2-7-8-3 all have cycle 4 (they are the tricky ones). Others are easy: 0,1,5,6 never change; 4 and 9 have cycle 2.
• HCF = Highest = Hurry and Find Common factors (go smaller). LCM = Lowest Multiple = Let Calculator Multiply (go bigger).
• Factor formula: POWER PLUS ONE then MULTIPLY ALL. Just add 1 to every exponent after prime factorization, then multiply them all.

• Agniveer papers typically have 1-2 LCM/HCF word problems phrased as bell-ringing or tiling problems — practice those specific templates.
• Unit digit questions appear almost every year — they take less than 30 seconds if you know cycles, so attempt them first.
• Divisibility rule for 11 is asked directly as MCQ: is this 5-digit number divisible by 11? Apply alternating sum rule immediately.
• Factor-counting questions are medium difficulty and appear 1-2 times — always prime factorize first before attempting.
• Avoid spending more than 60 seconds on any Number System question — all are solvable by rule or shortcut, never need long calculation.