Unit 1: Number System and Average

I. Orientation

The number system provides the language for counting, comparison, division, and measurement. In analytical reasoning, problems usually depend on the properties of integers, prime factorization, remainders, common multiples, and the interpretation of an average as a balanced central value.

• Natural numbers: Counting numbers beginning with (1): (1,2,3,\ldots). Some conventions include (0), so the context must be checked.
• Whole numbers: Natural numbers together with zero: (0,1,2,3,\ldots).
• Integers: Positive numbers, negative numbers, and zero: (\ldots,-2,-1,0,1,2,\ldots).
• Rational numbers: Numbers expressible as (\frac{p}{q}), where (p,q) are integers and (q\neq0), such as (\frac{3}{4}) and (-2=\frac{-2}{1}).
• Irrational numbers: Numbers that cannot be expressed as a ratio of integers, such as (\sqrt2) and (\pi).
• Real numbers: The union of rational and irrational numbers.
• Prime and composite numbers: A prime has exactly two positive factors, while a composite number has more than two. The number (1) is neither prime nor composite.
• Basic convention: Unless stated otherwise, factors, divisors, and HCF refer to positive integers.

II. Number Classification and Properties

Number classification identifies the type and structural properties of a number. These classifications support divisibility tests, factor counting, and calculations involving powers.

A. Classification of numbers

Classification of numbers is based on sign, representation, and the number of positive divisors.

• Prime numbers: A number greater than (1) with exactly two factors, (1) and itself. For example, (13) is prime because its only positive divisors are (1) and (13).
• Composite numbers: A number greater than (1) with at least one factor other than (1) and itself. For example, (18) is composite because (18=2\times3^2).
• Even and odd numbers: An even integer is divisible by (2), such as (28); an odd integer is not, such as (29).
  • Even-number form: (2k), where (k) is an integer.
  • Odd-number form: (2k+1), where (k) is an integer.
• Co-prime numbers: Two numbers whose HCF is (1), such as (8) and (15). They need not both be prime.
• Perfect squares and cubes: A perfect square has the form (n^2), such as (49=7^2); a perfect cube has the form (n^3), such as (125=5^3).
• Decimal classification: A rational number has a terminating decimal when its reduced denominator contains only (2) and/or (5). Thus, (\frac{7}{40}=0.175), while (\frac{1}{3}=0.333\ldots) is recurring.

B. Divisibility rules

Divisibility rules provide quick tests for whether one integer divides another without requiring complete division.

• Divisibility by 2: The last digit must be even: (0,2,4,6,) or (8). Hence (5,728) is divisible by (2).
• Divisibility by 3 and 9: The sum of digits must be divisible by (3) or (9), respectively. For (7,452), the digit sum is (18), so it is divisible by both.
• Divisibility by 4 and 8: The last two digits must be divisible by (4), and the last three digits by (8). Therefore, (1,316) is divisible by (4) because (16) is divisible by (4).
• Divisibility by 5 and 10: The last digit must be (0) or (5) for (5), and exactly (0) for (10).
• Divisibility by 6: The number must be divisible by both (2) and (3). Since (246) is even and (2+4+6=12), it is divisible by (6).
• Divisibility by 11: The difference between the sums of alternate digits must be (0) or a multiple of (11). For (1,452), ((1+5)-(4+2)=0), so it is divisible by (11).
• Divisibility by composite numbers: A test can be combined from factors. Divisibility by (12) requires divisibility by both (3) and (4), because (12=3\times4) and these factors are co-prime.

C. Factorials

A factorial is the product of all positive integers up to a specified non-negative integer.

• Definition: For (n\geq1),

n! = n × (n − 1) × (n − 2) × ... × 2 × 1

Here, (n) is a non-negative integer. For example, (5!=5\times4\times3\times2\times1=120).

• Special value: (0!=1). This convention makes formulas such as (\binom n0=\frac{n!}{0!n!}=1) valid.
• Recursive relation: (n!=n(n-1)!). Thus, (6!=6\times5!=720).
• Trailing zeroes: The number of zeroes at the end of (n!) equals the number of pairs of factors (2) and (5); factors of (5) are fewer.

Trailing zeroes in n! = floor(n/5) + floor(n/25) + floor(n/125) + ...

For (100!), the count is (20+4=24), because higher powers contribute zero.

D. Unit digit calculation

Unit digit calculation determines the last digit of an integer or power by using cyclic patterns and modular arithmetic.

• Powers of numbers ending in 2: The unit digits cycle as (2,4,8,6), with period (4). The exponent is reduced modulo (4).
• General method: For (a^n), calculate the repeating unit-digit cycle of (a), then use the position of (n) in that cycle.
• Worked example: To find the unit digit of (7^{103}), observe the cycle (7,9,3,1). Since (103\div4) leaves remainder (3), the unit digit is the third value, (3).
• Products: Only unit digits are needed. The unit digit of (38\times47) is the unit digit of (8\times7=56), namely (6).
• Special endings: Any positive power of a number ending in (0,1,5,) or (6) retains that same unit digit.

E. Number of factors

The number of positive factors can be found directly from the prime factorization of an integer.

• Prime factorization form: If

N = p₁^a × p₂^b × p₃^c

where the (p_i) are distinct primes, then the number of positive factors is

d(N) = (a + 1)(b + 1)(c + 1)

• Reason: A factor may contain (p_1) with exponent (0) through (a), giving (a+1) choices, independently for each prime.
• Worked example: (360=2^3\times3^2\times5^1), so

d(360) = (3 + 1)(2 + 1)(1 + 1) = 24

• Perfect-square property: A number has an odd number of factors exactly when it is a perfect square. For (36=2^2\times3^2), (d(36)=3\times3=9), which is odd.
• Factor pairs: Non-square numbers have factors in distinct pairs, while a square has one repeated middle pair, such as (6\times6=36).

F. Remainder theorem

The remainder theorem determines the remainder after division using a congruence or quotient-remainder representation.

• Division identity: For integers (a) and positive (b),

a = bq + r, where 0 ≤ r < b

Here, (q) is the quotient and (r) is the remainder.

• Congruence notation: (a\equiv r\pmod b) means (a) and (r) leave the same remainder when divided by (b).
• Polynomial form: If a polynomial (f(x)) is divided by (x-a), the remainder is (f(a)). For example, the remainder when (x^2+3x+5) is divided by (x-2) is (f(2)=4+6+5=15).
• Operations with remainders: If (a\equiv r_1\pmod m) and (b\equiv r_2\pmod m), then

a + b ≡ r₁ + r₂ (mod m)
ab ≡ r₁r₂ (mod m)

The resulting value can be reduced again to a remainder below (m).

• Negative remainder handling: A remainder must be non-negative under the standard convention. Thus, (-3\pmod7) has remainder (4).

III. HCF and LCM

HCF and LCM describe common divisibility relationships and are central to grouping, scheduling, and fraction calculations.

A. HCF and LCM

The HCF is the greatest common divisor; the LCM is the least positive common multiple.

• Prime-factor method: For (72=2^3\times3^2) and (120=2^3\times3\times5):
  1. HCF: Choose common primes with the smaller powers: (2^3\times3=24).
  2. LCM: Choose every prime with the larger powers: (2^3\times3^2\times5=360).
• Euclidean algorithm: Repeatedly divide and replace the pair by divisor and remainder.

HCF(a, b) = HCF(b, a mod b)

For (252) and (105): (252=2(105)+42), (105=2(42)+21), so the HCF is (21).

• Product relation: For positive integers (a,b),

HCF(a, b) × LCM(a, b) = a × b

This relation is valid for two numbers and provides a quick way to find one quantity when the other is known.

• Applications: HCF is used for largest equal grouping; LCM is used for the first simultaneous occurrence of repeating events, such as schedules of (12) and (18) days.

IV. Average and Aggregation

An average represents the equal-sharing value of a collection. It is calculated from a total and the number of observations.

A. Average calculations

Average calculations connect the total quantity to the number of items being averaged.

• Arithmetic mean: For observations (x_1,x_2,\ldots,x_n),

Average = (x₁ + x₂ + ... + xₙ) / n

Here, (n) is the number of observations. The average of (8,12,15) is (\frac{35}{3}=11\frac23).

• Total from average:

Total = Average × Number of observations

If the average of (18) values is (25), their total is (18\times25=450).

• Change in average: If one value changes by (d), the total changes by (d), so the average changes by (\frac dn). Replacing (40) by (52) among (6) values raises the average by (\frac{12}{6}=2).
• Combined average: For groups with totals (T_1,T_2) and sizes (n_1,n_2),

Combined average = (T₁ + T₂) / (n₁ + n₂)

Group averages must not be combined by simple averaging unless the group sizes are equal.

B. Inclusion and exclusion

Inclusion and exclusion prevents double-counting when sets overlap.

• Two-set formula: For sets (A) and (B),

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Here, (A\cup B) means at least one set, and (A\cap B) means both sets.

• Why subtraction is necessary: An item belonging to both sets is counted once in (n(A)) and once in (n(B)), so it must be subtracted once.
• Complement form: If a universal set has (N) elements, the number belonging to neither set is

N − n(A ∪ B)

• Three-set formula:

n(A ∪ B ∪ C)
= n(A)+n(B)+n(C)
−n(A∩B)−n(B∩C)−n(C∩A)
+n(A∩B∩C)

The triple intersection is added because it was subtracted too many times.

C. Weighted average

A weighted average gives observations different importance according to their assigned weights.

• Formula: If values (x_1,x_2,\ldots,x_n) have weights (w_1,w_2,\ldots,w_n),

Weighted average = (w₁x₁ + w₂x₂ + ... + wₙxₙ) / (w₁ + w₂ + ... + wₙ)

The (w_i) represent frequency, quantity, credit, or importance.

• Frequency interpretation: If scores (70) and (85) occur (2) and (3) times, the weighted average is

(2×70 + 3×85) / (2+3) = 395/5 = 79

• Weighted versus ordinary average: The ordinary average treats every observation equally; the weighted average gives greater influence to the value with the larger weight.
• Practical application: A course grade with assignments worth (40\%) and an examination worth (60\%) is calculated as (0.40A+0.60E), where (A) and (E) are the two scores.
• Consistency condition: Weights should be non-negative and measured on a compatible basis. If weights are percentages, they should normally total (100\%); if they are ratios, only their relative sizes matter.