Civil Service Exam (Subprofessional) Numerical Ability — Integers, PEMDAS & DivisibilityRevision Notes

Formulas

Exam Tips

• Remember: Zero has special properties - neither positive/negative, neither prime/composite
• When working with absolute values, always make the result positive
• Practice the law of signs for all four operations
• Memorize that any number to the power of zero equals one (except 0^0)

Key Points

• Integers include all positive numbers, negative numbers, and zero: {...-3, -2, -1, 0, 1, 2, 3...}
• Zero is neither positive nor negative, but it is an even number
• Zero is neither prime nor composite
• Positive integers are also called natural numbers or counting numbers
• Negative integers are numbers preceded by a negative sign (-)
• Every integer has an additive inverse (opposite): 5 and -5 are additive inverses
• Absolute value |n| represents the distance from zero on the number line

Definitions

Section Title

Integers and Their Properties

Common Mistakes

• Thinking zero is positive or negative - zero is neutral
• Confusing the absolute value of negative numbers - |-5| = 5, not -5
• Forgetting that zero divided by any non-zero number equals zero
• Attempting to divide by zero - this is undefined

Formulas

Exam Tips

• Create a quick reference: Same signs = positive, Different signs = negative (for × and ÷)
• For addition with different signs, subtract smaller from larger and use sign of larger
• Practice converting subtraction to addition: 8 - (-3) = 8 + 3 = 11
• Double-check your signs - this is where most errors occur

Key Points

• Addition: Same signs → add and keep the sign; Different signs → subtract and use sign of larger absolute value
• Subtraction: Change to addition of the opposite → a - b = a + (-b)
• Multiplication: Same signs → positive result; Different signs → negative result
• Division: Same signs → positive result; Different signs → negative result
• Remember: (+)(+) = +, (-)(-) = +, (+)(-) = -, (-)(+) = -

Definitions

Section Title

Law of Signs for Integer Operations

Common Mistakes

• Getting confused with subtraction - remember to change to addition of opposite
• Forgetting that two negatives make a positive in multiplication/division
• Mixing up addition and multiplication rules for signs
• Not applying the larger absolute value rule correctly in addition

Formulas

Exam Tips

• Always start with parentheses - work from inside out
• Remember that M&D have equal priority, as do A&S - go left to right
• Write out each step clearly to avoid errors
• When in doubt, use extra parentheses to clarify your intended order

Key Points

• PEMDAS stands for: Parentheses, Exponents, Multiplication & Division, Addition & Subtraction
• Always work from innermost parentheses outward: ( ) then [ ] then { }
• Exponents are calculated from outermost inward: 3^(2^3) = 3^8 = 6561
• Multiplication and Division have equal priority - work left to right
• Addition and Subtraction have equal priority - work left to right
• Number next to parentheses means multiplication: 3(4) = 3 × 4

Definitions

Section Title

PEMDAS - Order of Operations

Common Mistakes

• Working strictly left to right without considering operation priority
• Forgetting to evaluate what's inside parentheses first
• Not recognizing implied multiplication: 2(3 + 4) means 2 × (3 + 4)
• Confusing the order of exponents: 2^3^2 = 2^(3^2) = 2^9, not (2^3)^2

Formulas

Exam Tips

• List factors in pairs: for 12, think (1,12), (2,6), (3,4)
• Use prime factorization for finding GCF and LCM efficiently
• Remember: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29... are the first primes
• For GCF, multiply common prime factors; for LCM, use all prime factors with highest powers

Key Points

• Factors are numbers that divide evenly into another number
• Every number has 1 and itself as factors
• Prime numbers have exactly two factors: 1 and themselves
• Composite numbers have more than two factors
• Multiples are products of a number and positive integers
• Greatest Common Factor (GCF) is the largest factor shared by two or more numbers
• Least Common Multiple (LCM) is the smallest multiple shared by two or more numbers

Definitions

Section Title

Factors and Multiples

Common Mistakes

• Forgetting that 1 is neither prime nor composite
• Confusing factors and multiples - factors divide into, multiples are divided by
• Missing factors when listing - be systematic
• Thinking 2 is not prime because it's even - 2 is the only even prime

Formulas

Exam Tips

• Memorize the most common rules: 2, 3, 5, 9, 10
• For 3 and 9: If sum of digits has many digits, repeat the process
• Practice the alternating sum for rule of 11
• Use these rules to eliminate wrong answers quickly in multiple choice

Key Points

• Divisibility by 2: Last digit is even (0, 2, 4, 6, 8)
• Divisibility by 3: Sum of digits is divisible by 3
• Divisibility by 4: Last two digits form a number divisible by 4
• Divisibility by 5: Last digit is 0 or 5
• Divisibility by 6: Divisible by both 2 and 3
• Divisibility by 8: Last three digits form a number divisible by 8
• Divisibility by 9: Sum of digits is divisible by 9
• Divisibility by 10: Last digit is 0
• Divisibility by 11: Alternating sum of digits is divisible by 11

Definitions

Section Title

Divisibility Rules

Common Mistakes

• Mixing up the rules - memorize each one separately
• For rule of 11, counting positions incorrectly
• Forgetting that divisibility by 6 requires BOTH 2 and 3 conditions
• Not checking last 2 or 3 digits correctly for rules of 4 and 8

Formulas

Exam Tips

• Count decimal places moved to determine exponent
• Moving right = positive exponent, moving left = negative exponent
• Check: large numbers have positive exponents, small numbers have negative
• Practice converting both ways: decimal to scientific and scientific to decimal

Key Points

• Scientific notation: m × 10^n where 1 ≤ |m| < 10 and n is an integer
• Positive exponent: decimal moves right (large numbers)
• Negative exponent: decimal moves left (small numbers)
• Zero has special properties in all operations
• Any number to the power of zero equals 1 (except 0^0 which is indeterminate)
• Scientific notation simplifies very large or very small numbers

Definitions

Section Title

Scientific Notation and Special Numbers

Common Mistakes

• Getting the decimal point movement direction wrong
• Not ensuring the coefficient is between 1 and 10
• Confusing positive and negative exponents
• Forgetting that 10^0 = 1