Civil Service Exam (Professional) Numerical Ability — Integers, PEMDAS & DivisibilityExam Answer Templates

Examiner tip

Include the phrase 'cannot be a fraction' to show complete understanding

Model answer

An integer is a whole number that can be positive, negative, or zero, but cannot be a fraction or decimal.

Examples: -5, 0, 7 (any two appropriate examples)

Answer structure

• Line 1: Define integer clearly [1 mark]
• Line 2: Provide two correct examples [1 mark]

Scoring breakdown

Common mark deductions

• Incomplete definition missing key elements
• Giving fractional or decimal examples
• Only one example provided

Key phrases to include

• whole number
• positive, negative, or zero
• cannot be a fraction

Examiner tip

Always convert double negatives to positive immediately to avoid confusion

Model answer

(-8) + (+3) - (-5)
= (-8) + (+3) + (+5) [Converting subtraction of negative to addition]
= -8 + 3 + 5
= -8 + 8 = 0

Answer: 0

Answer structure

• Line 1: Write the original expression [no marks, but essential]
• Line 2: Apply sign rules correctly [1 mark]
• Line 3: Perform addition/subtraction step by step [1 mark]
• Line 4: State final answer clearly [1 mark]

Scoring breakdown

Common mark deductions

• Sign errors when handling negative numbers
• Not showing intermediate steps
• Arithmetic mistakes

Key phrases to include

• subtracting negative equals adding positive
• step-by-step working

Examiner tip

Label each PEMDAS step to show you understand the correct order

Model answer

2 + 3 × (4 - 1)²

Step 1 - Parentheses: 4 - 1 = 3
2 + 3 × (3)²

Step 2 - Exponents: (3)² = 9
2 + 3 × 9

Step 3 - Multiplication: 3 × 9 = 27
2 + 27

Step 4 - Addition: 2 + 27 = 29

Answer: 29

Answer structure

• Line 1: Identify and solve parentheses [1 mark]
• Line 2: Apply exponents [1 mark]
• Line 3: Perform multiplication and addition [1 mark]

Scoring breakdown

Common mark deductions

• Wrong order of operations
• Calculation errors
• Not labeling each step clearly

Key phrases to include

• PEMDAS order
• parentheses first
• exponents second
• multiplication before addition

Examiner tip

Always state the rule first, then apply it systematically

Model answer

Rule for divisibility by 8: A number is divisible by 8 if its last three digits form a number divisible by 8.

Last three digits of 2,376 = 376
376 ÷ 8 = 47 (exactly)

Therefore, 2,376 is divisible by 8.

Answer structure

• Line 1: State the divisibility rule for 8 [1 mark]
• Line 2: Apply the rule and show calculation [1 mark]

Scoring breakdown

Common mark deductions

• Not stating the rule
• Wrong rule application
• Calculation errors

Key phrases to include

• last three digits
• divisibility rule
• exactly divisible

Examiner tip

Show every single step and state which PEMDAS rule you're applying at each stage

Model answer

48 ÷ 6 + 2³ × (10 - 7) - 5

Step 1 - Parentheses first:
(10 - 7) = 3
Expression becomes: 48 ÷ 6 + 2³ × 3 - 5

Step 2 - Exponents:
2³ = 2 × 2 × 2 = 8
Expression becomes: 48 ÷ 6 + 8 × 3 - 5

Step 3 - Multiplication and Division (left to right):
48 ÷ 6 = 8
8 × 3 = 24
Expression becomes: 8 + 24 - 5

Step 4 - Addition and Subtraction (left to right):
8 + 24 = 32
32 - 5 = 27

Final Answer: 27

Answer structure

• Step 1: Handle parentheses [1 mark]
• Step 2: Calculate exponents [1 mark]
• Step 3: Perform multiplication and division [2 marks]
• Step 4: Complete addition and subtraction [1 mark]

Scoring breakdown

Common mark deductions

• Wrong order of operations
• Calculation errors in exponents
• Not working left to right for same-level operations
• Missing intermediate steps

Key phrases to include

• PEMDAS order
• parentheses first
• left to right
• step by step

Examiner tip

Grouping like signs first makes the problem much easier and reduces errors

Model answer

(-12) + (+8) + (-15) + (+20)

Group positive and negative integers:
Positive integers: +8 + 20 = +28
Negative integers: (-12) + (-15) = -27

Combine results:
+28 + (-27) = +28 - 27 = +1

Answer: +1 or 1

Answer structure

• Line 1: Group like signs together [1 mark]
• Line 2: Add within each group [1 mark]
• Line 3: Combine final results [1 mark]

Scoring breakdown

Common mark deductions

• Sign errors
• Not grouping systematically
• Final combination errors

Key phrases to include

• group like signs
• add positive integers
• add negative integers

Examiner tip

Always test both conditions (divisibility by 2 AND 3) to show complete understanding

Model answer

Divisibility rule for 6: A number is divisible by 6 if it is divisible by both 2 and 3.

Testing 234:
For divisibility by 2: Last digit is 4 (even), so 234 is divisible by 2 ✓
For divisibility by 3: Sum of digits = 2 + 3 + 4 = 9, and 9 ÷ 3 = 3, so 234 is divisible by 3 ✓

Since 234 is divisible by both 2 and 3, it is divisible by 6.

Answer structure

• Line 1: State divisibility rule for 6 [1 mark]
• Line 2: Test divisibility by 2 [1 mark]
• Line 3: Test divisibility by 3 and conclude [1 mark]

Scoring breakdown

Common mark deductions

• Incomplete rule statement
• Wrong divisibility tests
• Not testing both conditions

Key phrases to include

• divisible by both 2 and 3
• even number
• sum of digits

Examiner tip

Use proper absolute value notation with the vertical bars

Model answer

|-25| = 25

Answer structure

• Direct answer with proper notation [1 mark]

Scoring breakdown

Common mark deductions

• Wrong notation
• Negative answer
• Missing absolute value bars

Key phrases to include

• absolute value notation | |
• positive value

Examiner tip

State the sign rule you're using at each step to show understanding

Model answer

(-6) × (+4) × (-2)

Apply multiplication rules for integers:
Negative × Positive = Negative: (-6) × (+4) = -24
Negative × Negative = Positive: (-24) × (-2) = +48

Answer: +48 or 48

Answer structure

• Line 1: Apply sign rules step by step [1 mark]
• Line 2: Calculate final answer [1 mark]

Scoring breakdown

Common mark deductions

• Sign errors
• Wrong multiplication rules
• Calculation errors

Key phrases to include

• sign rules
• negative × positive = negative
• negative × negative = positive

Examiner tip

Double-check your addition of digits as this is where most errors occur

Model answer

Divisibility rule for 9: A number is divisible by 9 if the sum of its digits is divisible by 9.

Sum of digits of 5,427: 5 + 4 + 2 + 7 = 18
18 ÷ 9 = 2 (exactly)

Therefore, 5,427 is divisible by 9.

Answer structure

• Line 1: State the divisibility rule for 9 [1 mark]
• Line 2: Apply rule and conclude [1 mark]

Scoring breakdown

Common mark deductions

• Wrong rule
• Addition errors
• Incorrect conclusion

Key phrases to include

• sum of digits
• divisible by 9
• divisibility rule

Examiner tip

Remember that multiplication and division have equal priority - work left to right

Model answer

15 - 3 × 2² + 8 ÷ 4

Step 1 - Exponents:
2² = 4
Expression becomes: 15 - 3 × 4 + 8 ÷ 4

Step 2 - Multiplication and Division (left to right):
3 × 4 = 12
8 ÷ 4 = 2
Expression becomes: 15 - 12 + 2

Step 3 - Addition and Subtraction (left to right):
15 - 12 = 3
3 + 2 = 5

Final Answer: 5

Answer structure

• Step 1: Handle exponents [1 mark]
• Step 2: Perform multiplication and division [2 marks]
• Step 3: Complete addition and subtraction [1 mark]

Scoring breakdown

Common mark deductions

• Wrong order of operations
• Exponent calculation errors
• Not working left to right

Key phrases to include

• PEMDAS order
• exponents first
• left to right

Examiner tip

Be specific about which digits qualify as 'even' to avoid ambiguity

Model answer

Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).

Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.

Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.

Answer structure

• Line 1: Rule for divisibility by 2 [1 mark]
• Line 2: Rule for divisibility by 3 [1 mark]
• Line 3: Rule for divisibility by 5 [1 mark]

Scoring breakdown

Common mark deductions

• Incomplete rules
• Mixing up rules
• Not mentioning all required digits

Key phrases to include

• even digit
• sum of digits
• ends in 0 or 5

Examiner tip

Remember: moving decimal right gives negative exponent, left gives positive

Model answer

0.00456

Move decimal point to create a number between 1 and 10:
0.00456 → 4.56
Decimal moved 3 places to the right, so exponent is -3

Scientific notation: 4.56 × 10⁻³

Answer structure

• Line 1: Identify decimal movement [1 mark]
• Line 2: Write in proper scientific notation [1 mark]

Scoring breakdown

Common mark deductions

• Wrong exponent sign
• Incorrect decimal placement
• Wrong format

Key phrases to include

• move decimal point
• between 1 and 10
• negative exponent

Examiner tip

Remember absolute value always gives the positive distance from zero

Model answer

|−8| + |+5| − |−3|

Evaluate each absolute value:
|−8| = 8
|+5| = 5
|−3| = 3

Substitute and calculate:
8 + 5 − 3 = 13 − 3 = 10

Answer: 10

Answer structure

• Line 1: Evaluate each absolute value [1 mark]
• Line 2: Perform final calculation [1 mark]

Scoring breakdown

Common mark deductions

• Wrong absolute values
• Sign errors in final calculation
• Arithmetic mistakes

Key phrases to include

• absolute value
• positive value
• distance from zero

Examiner tip

Always mention that division BY zero is undefined - this is a key distinction

Model answer

Zero Property in Multiplication:
Any number multiplied by zero equals zero.
Example: 7 × 0 = 0, (-5) × 0 = 0

Zero Property in Division:
Zero divided by any non-zero number equals zero, but division by zero is undefined.
Example: 0 ÷ 8 = 0, but 8 ÷ 0 is undefined

Answer structure

• Line 1: State multiplication property with example [1 mark]
• Line 2: State division property [1 mark]
• Line 3: Provide correct examples [1 mark]

Scoring breakdown

Common mark deductions

• Missing the undefined division case
• Wrong examples
• Incomplete explanations

Key phrases to include

• any number times zero equals zero
• zero divided by non-zero equals zero
• division by zero is undefined

Do

• Be precise and to the point
• Use correct mathematical notation
• Circle your final answer

Don't

• Don't over-explain
• Don't show unnecessary working
• Don't leave answers unclear

Do

• State rules before applying them
• Show key working step
• Label your answer clearly

Don't

• Don't skip the rule statement
• Don't make calculation errors
• Don't provide incomplete answers

Do

• Number your steps clearly
• Show intermediate calculations
• State which rule you're using

Don't

• Don't jump steps
• Don't make sign errors
• Don't leave calculations unfinished

Do

• Show every calculation step
• Explain your reasoning
• Use proper PEMDAS labels
• Double-check your work

Don't

• Don't rush through steps
• Don't make order of operations errors
• Don't skip intermediate results
• Don't forget to state your final answer clearly