4.1 Arithmetic Operations, Number Properties & Order of Operations

Arithmetic Operations, Number Properties & Order of Operations

Military command demands rapid, error-free quantitative estimation and precise numerical calculation. Whether computing ammunition consumption rates, establishing ballistic mortar ranges, rationing logistical fuel reserves, or allocating manpower across tactical operational sectors, an officer must possess absolute fluency in foundational arithmetic. On the Ghana Armed Forces (GAF) Officer Cadet Written Examination, the quantitative aptitude section assesses speed, analytical rigor, and conceptual depth under strict time constraints.

Mastery begins with a rigorous comprehension of number properties, algebraic structures, divisibility mechanics, and order of operations. Candidates who rely on slow, manual long division or ambiguous operational hierarchies inevitably suffer from cognitive fatigue and test pacing failures.

Classification of Numbers & Structural Taxonomy

The real number system, denoted by $\mathbb{R}$, forms the operational space for military quantitative aptitude. Candidates must recognize the formal hierarchy and boundaries between distinct numerical classifications.

                             REAL NUMBERS (R)
                                    |
         +--------------------------+--------------------------+
         |                                                     |
  RATIONAL NUMBERS (Q)                                  IRRATIONAL NUMBERS
  (Terminating & Recurring Decimals)                    (Non-terminating, Non-recurring:
         |                                               sqrt(2), sqrt(3), pi, e)
         +--------------------------+
         |                          |
    FRACTIONS                   INTEGERS (Z)
    (Proper, Improper, Mixed)   (..., -2, -1, 0, 1, 2, ...)
                                    |
                                    +--------------------------+
                                    |                          |
                             NEGATIVE INTEGERS          WHOLE NUMBERS (W)
                             (-1, -2, -3, ...)          (0, 1, 2, 3, ...)
                                                               |
                                                +--------------+--------------+
                                                |                             |
                                              ZERO (0)                 NATURAL NUMBERS (N)
                                                                       (1, 2, 3, 4, ...)
                                                                              |
                                                                +-------------+-------------+
                                                                |             |             |
                                                              PRIME       COMPOSITE       UNIT
                                                             (2,3,5..)    (4,6,8,9..)      (1)

Formal Definitions and Operational Distinctions

Properties of Prime, Composite, and Coprime Numbers

• Prime Numbers: A natural number $p > 1$ that possesses exactly two distinct positive divisors: $1$ and $p$. The first ten primes are $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$. Note that $2$ is the unique even prime number and the smallest prime.
• Composite Numbers: A natural number $n > 1$ that possesses more than two distinct positive divisors (e.g., $4, 6, 8, 9, 10, 12$).
• The Unit ($1$): The integer $1$ has only one positive divisor ($1$ itself). Therefore, $1$ is strictly neither prime nor composite.
• Coprime (Relatively Prime) Integers: Two integers $a$ and $b$ are coprime if and only if their greatest common divisor is $1$, written as $\gcd(a, b) = 1$. For example, $8$ and $15$ are both composite, yet $\gcd(8, 15) = 1$, making them coprime.

The Prime Verification Test: To determine whether an integer $N$ is prime, identify the largest integer $k$ such that $k \le \sqrt{N}$. Test $N$ for divisibility only by the prime numbers less than or equal to $k$. If no prime $p \le \sqrt{N}$ divides $N$, then $N$ is unconditionally prime.

Example: Test whether $173$ is prime. Since $\sqrt{173} \approx 13.15$, test primes $p \le 13$: $\{2, 3, 5, 7, 11, 13\}$. None divide $173$ evenly ($173 = 7 \times 24 + 5 = 11 \times 15 + 8 = 13 \times 13 + 4$). Thus, $173$ is prime.

Comprehensive Divisibility Rules & Modular Verification

Divisibility rules allow an officer to instantly verify supply counts, crate packaging, and modular groupings without executing tedious paper-and-pencil long division.

Divisibility Protocols (2 Through 12)

The Coprime Factor Theorem for Composite Divisors

To test divisibility by any composite divisor $C$, express $C$ as the product of two coprime integers $a$ and $b$ such that $C = a \times b$ and $\gcd(a, b) = 1$. The number $N$ is divisible by $C$ if and only if it is divisible by both $a$ and $b$.

• Divisibility by $15$: Test divisibility by $3$ and $5$ (since $\gcd(3, 5) = 1$).
• Divisibility by $18$: Test divisibility by $2$ and $9$ (since $\gcd(2, 9) = 1$).
• Divisibility by $72$: Test divisibility by $8$ and $9$ (since $\gcd(8, 9) = 1$).
• Warning: Do not use non-coprime factors! Testing divisibility by $2$ and $6$ does NOT prove divisibility by $12$, because $18$ is divisible by both $2$ and $6$, yet $18$ is not divisible by $12$.

LCM, HCF (GCD), and the Euclidean Algorithm

Logistics planning frequently requires finding the greatest common unit size to pack heterogeneous assets (Highest Common Factor / HCF) or calculating the synchronization cycle of independent periodic resupply convoys (Lowest Common Multiple / LCM).

1. Prime Factorization Method

Let two positive integers be resolved into their canonical prime factorizations: $a = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}, \quad b = p_1^{f_1} p_2^{f_2} \cdots p_k^{f_k}$

• Highest Common Factor (HCF / GCD): Take the lowest exponent for every common prime factor: $\operatorname{HCF}(a, b) = p_1^{\min(e_1, f_1)} p_2^{\min(e_2, f_2)} \cdots p_k^{\min(e_k, f_k)}$
• Lowest Common Multiple (LCM): Take the highest exponent across all prime factors present in either number: $\operatorname{LCM}(a, b) = p_1^{\max(e_1, f_1)} p_2^{\max(e_2, f_2)} \cdots p_k^{\max(e_k, f_k)}$

2. The Fundamental Product Invariant

For any two positive integers $a$ and $b$, the product of their HCF and LCM is identically equal to the product of the numbers themselves: $\operatorname{HCF}(a, b) \times \operatorname{LCM}(a, b) = a \times b$

This identity provides a rapid verification tool and allows calculating the LCM directly once the HCF is known: $\operatorname{LCM}(a, b) = \frac{a \times b}{\operatorname{HCF}(a, b)}$.

3. The Euclidean Algorithm for Rapid HCF Computation

When dealing with large integers (e.g., ammunition batches of $1,848$ and $1,155$), prime factorization becomes slow and prone to arithmetic blunders. The Euclidean algorithm operates on the principle that the greatest common divisor of two integers $a$ and $b$ ($a > b$) also divides their remainder $r = a \pmod b$: $\operatorname{GCD}(a, b) = \operatorname{GCD}(b, a \pmod b)$

Worked Example: Calculating $\operatorname{GCD}(1848, 1155)$

1. Step 1: Divide $1848$ by $1155$: $1848 = 1155 \times 1 + 693$
2. Step 2: Divide the previous divisor ($1155$) by the remainder ($693$): $1155 = 693 \times 1 + 462$
3. Step 3: Divide $693$ by $462$: $693 = 462 \times 1 + 231$
4. Step 4: Divide $462$ by $231$: $462 = 231 \times 2 + 0$

Because the remainder has reached $0$, the final non-zero divisor is the greatest common factor: $\operatorname{HCF}(1848, 1155) = 231$

To find their LCM using the product invariant: $\operatorname{LCM}(1848, 1155) = \frac{1848 \times 1155}{231} = 1848 \times 5 = 9,240$

4. Operational Application: Ammunition Packaging

Suppose a quartermaster at Burma Camp receives three consignments of rifle cartridges: $315$ rounds of $7.62\text{ mm}$, $420$ rounds of $5.56\text{ mm}$, and $1,155$ rounds of $9\text{ mm}$. The ammunition must be packed into identical crates such that each crate contains the exact same composition of calibers with zero ammunition remaining unpacked.

• The maximum number of identical crates possible is given by $\operatorname{HCF}(315, 420, 1155)$:
  • $315 = 3^2 \times 5 \times 7$
  • $420 = 2^2 \times 3 \times 5 \times 7$
  • $1155 = 3 \times 5 \times 7 \times 11$
  • Common prime factors: $3^1, 5^1, 7^1$
  • $\operatorname{HCF} = 3 \times 5 \times 7 = 105$ crates.
• Each crate contains: $\frac{315}{105} = 3$ rounds ($7.62\text{ mm}$), $\frac{420}{105} = 4$ rounds ($5.56\text{ mm}$), and $\frac{1155}{105} = 11$ rounds ($9\text{ mm}$).

BODMAS / PEMDAS Hierarchies & Nested Traps

The order of operations defines the unambiguous sequence in which mathematical expressions must be evaluated. In Ghanaian schools, as across the Commonwealth, the BODMAS acronym is standard (equivalent to the American PEMDAS).

B - Brackets (Vinculum, Parentheses, Braces, Square Brackets)
O - Orders (Powers, Indices, Roots)
D - Division       [Equal Priority: Executed Strictly Left-to-Right]
M - Multiplication [Equal Priority: Executed Strictly Left-to-Right]
A - Addition       [Equal Priority: Executed Strictly Left-to-Right]
S - Subtraction    [Equal Priority: Executed Strictly Left-to-Right]

The Left-to-Right Precedence Rule

A pervasive error on officer aptitude tests is treating Division as strictly superior to Multiplication, or Addition as strictly superior to Subtraction. In formal mathematics:

• Division and Multiplication share identical operational priority. When both appear in a chain, evaluate them strictly in order from left to right.
  • Example: $24 \div 6 \times 2$
  • Incorrect (premature multiplication): $24 \div (6 \times 2) = 24 \div 12 = 2$.
  • Correct (left-to-right evaluation): $(24 \div 6) \times 2 = 4 \times 2 = 8$.
• Addition and Subtraction share identical operational priority. Evaluate strictly from left to right.
  • Example: $15 - 8 + 4$
  • Incorrect (premature addition): $15 - (8 + 4) = 15 - 12 = 3$.
  • Correct (left-to-right evaluation): $(15 - 8) + 4 = 7 + 4 = 11$.

Hierarchy of Parenthetical Nesting

When multiple grouping symbols appear, evaluate from the innermost layer to the outermost layer in this sequence:

1. Vinculum (Bar / Overline): $\overline{a - b}$ (binds terms tighter than standard parentheses)
2. Parentheses (Round Brackets): $( \dots )$
3. Braces (Curly Brackets): $\{ \dots \}$
4. Square Brackets: $[ \dots ]$

Negative Sign Distribution Trap

A negative sign preceding a parenthetical expression distributes across every internal term, reversing its sign: $-[a - b + c] = -a + b - c$

Comprehensive Worked BODMAS Problem

Evaluate the following complex expression: $E = 72 \div 8 \times 3 - [24 - \{16 - 3 \times (10 - \overline{7 - 3})\}]$

• Step 1 (Resolve the Vinculum): The bar rests over $7 - 3$: $\overline{7 - 3} = 4$ Expression becomes: $72 \div 8 \times 3 - [24 - \{16 - 3 \times (10 - 4)\}]$
• Step 2 (Resolve Parentheses): Evaluate $10 - 4$: $10 - 4 = 6$ Expression becomes: $72 \div 8 \times 3 - [24 - \{16 - 3 \times 6\}]$
• Step 3 (Resolve Multiplication inside Braces): Multiply $3 \times 6$: $3 \times 6 = 18$ Expression becomes: $72 \div 8 \times 3 - [24 - \{16 - 18\}]$
• Step 4 (Resolve Braces): Evaluate $16 - 18$: $16 - 18 = -2$ Expression becomes: $72 \div 8 \times 3 - [24 - (-2)]$
• Step 5 (Resolve Square Brackets): Distribute the negative sign: $24 - (-2) = 24 + 2 = 26$: $[24 - (-2)] = 26$ Expression becomes: $72 \div 8 \times 3 - 26$
• Step 6 (Resolve Division and Multiplication Left-to-Right): $72 \div 8 = 9, \quad 9 \times 3 = 27$
• Step 7 (Final Subtraction): $27 - 26 = 1$

The exact simplified value is $1$.

Fractions, Decimals & Mental Math Heuristics

1. Terminating vs. Recurring Decimals

A rational fraction $\frac{p}{q}$ in irreducible form (where $\gcd(p, q) = 1$) converts to a terminating decimal if and only if the prime factorization of its denominator $q$ contains no prime factors other than $2$ and/or $5$ ($q = 2^m \times 5^n$). If $q$ contains any prime factor other than $2$ or $5$ (such as $3, 7, 11, 13$), the fraction generates an infinite repeating (recurring) decimal.

• $\frac{7}{40} = \frac{7}{2^3 \times 5^1} \implies$ Terminates ($0.175$).
• $\frac{5}{24} = \frac{5}{2^3 \times 3^1} \implies$ Recurs indefinitely ($0.208333\dots$).

2. Converting Repeating Decimals to Fractions

• Pure Recurring Decimals: All digits after the decimal point repeat. $0.\overline{d_1 d_2 \dots d_k} = \frac{d_1 d_2 \dots d_k}{\underbrace{99\dots9}_{k \text{ nines}}}$ Example: $0.\overline{45} = \frac{45}{99} = \frac{5}{11}$.
• Mixed Recurring Decimals: Some non-repeating digits precede the repeating block. $0.a_1 a_2 \dots a_m \overline{d_1 d_2 \dots d_k} = \frac{(\text{Entire Number up to end of first period}) - (\text{Non-repeating Part})}{\underbrace{99\dots9}_{k \text{ nines}} \underbrace{00\dots0}_{m \text{ zeros}}}$ Example: Convert $0.23\overline{6}$ to an irreducible fraction: $x = 0.23666\dots \implies 100x = 23.666\dots, \quad 1000x = 236.666\dots$ $1000x - 100x = 236 - 23 \implies 900x = 213$ $x = \frac{213}{900} = \frac{71}{300}$

3. Rapid Mental Math Shortcuts & Elimination Heuristics

• Units Digit Periodicity of Powers: In multiple-choice questions involving large exponents, compute only the cycle of the units digit modulo $4$:
  • Digits $\{0, 1, 5, 6\}$ always repeat every power (period $1$): $6^n$ always ends in $6$.
  • Digits $\{4, 9\}$ alternate in cycles of $2$: $4^1 = 4, 4^2 = 16, 4^3 = 64$ (odd powers end in $4$, even powers end in $6$).
  • Digits $\{2, 3, 7, 8\}$ cycle through $4$ values:
    • $2^1=2, 2^2=4, 2^3=8, 2^4=16 \implies$ Cycle: $\{2, 4, 8, 6\}$.
    • $3^1=3, 3^2=9, 3^3=27, 3^4=81 \implies$ Cycle: $\{3, 9, 7, 1\}$.
    • $7^1=7, 7^2=49, 7^3=343, 7^4=2401 \implies$ Cycle: $\{7, 9, 3, 1\}$.
• Digital Root (Casting Out Nines): The digital root of a number is the single-digit sum obtained by repeatedly adding its digits. The digital root of a sum or product must match the digital root of the inputs: $\operatorname{DR}(A \times B) = \operatorname{DR}(\operatorname{DR}(A) \times \operatorname{DR}(B))$ This provides instant verification of multi-digit arithmetic options.
• Parity Arithmetic: Leverage odd/even invariants to eliminate impossible answer choices: $\text{Even} \pm \text{Even} = \text{Even}, \quad \text{Odd} \pm \text{Odd} = \text{Even}, \quad \text{Even} \pm \text{Odd} = \text{Odd}$ $\text{Even} \times \text{Any} = \text{Even}, \quad \text{Odd} \times \text{Odd} = \text{Odd}$