Number System and Mensuration — AAI ATC GK Lesson

Before you start

Learning objectives and outcomes

Objective 1Distinguish whole numbers from natural numbers and apply it correctly.

Objective 2Use the unit-digit cycling rule to solve power questions quickly.

Objective 3Break compound divisibility checks into simpler factors.

Outcome ARecall every core mensuration formula and apply it directly, including for composite solids.

Subtopic 1

Number system essentials

This section covers the properties of whole and natural numbers, HCF/LCM, divisibility, and the recurring patterns in powers and factors.

Key distinctions and rules

Whole numbers start at 0 · Natural numbers start at 1
Unit digits of powers cycle in blocks of 4 · For a number = pᴮ×qᴭ, the number of factors = (a+1)(b+1)

The whole-versus-natural distinction is a favourite trap: since whole numbers include zero, the product of the smallest 8 whole numbers is always 0, not the same as the product of the smallest 8 natural numbers.

✈ AAI ATC relevance — the whole-vs-natural-number distinction and the unit-digit cycling rule are both tested directly, almost every attempt

Memory trick — "Zero ruins the product, every time"

Any product that includes 0 as a factor is always 0 — a single fact that instantly resolves the "smallest 8 whole numbers" style of question.

Easy

What is the product of the smallest 8 whole numbers?

Whole numbers begin at 0, so the smallest eight are 0 to 7, and any product containing 0 is itself 0.

Moderate

The unit digit of 7⁴⁵ is:

Unit digits of powers of 7 cycle every four steps: 7,9,3,1. Since 45 divided by 4 leaves remainder 1, the unit digit matches the first term of the cycle, 7.

Subtopic 2

Divisibility problems

Divisibility questions often combine two rules at once — for example, divisibility by 90 requires checking both divisibility by 9 and by 10 simultaneously.

Combined divisibility checks

Divisible by 10 → last digit is 0
Divisible by 9 → digit sum is divisible by 9
Divisible by 90 → both conditions must hold together

Breaking a compound divisor like 90 into its simpler factors (9 and 10) and checking each separately is far more reliable than trying to test divisibility by 90 directly.

Memory trick — "Split the divisor before you test it"

Whenever the divisor isn't a single simple number (like 90, or 72, or 45), break it into prime-ish factors first, and test each condition separately.

Hard

If x and y are the two digits of the number 3212xy such that the number is divisible by 90, then x+y equals:

Divisibility by 90 needs both by-10 (so y=0) and by-9 (digit sum 3+2+1+2+x+0=8+x divisible by 9, so x=1). Hence x+y=1.

Subtopic 3 · highest yield

Mensuration formulas

Mensuration questions are almost entirely direct formula substitution — the only real skill is remembering the right formula and not mixing up perimeter, area and volume.

Core mensuration formulas

Rectangle area = l×b · Triangle area = ½×base×height
Circle: circumference = 2πr, area = πr²
Cube: volume = a³, surface area = 6a² · Cylinder volume = πr²h
Trapezium area = ½×(sum of parallel sides)×height

For a composite solid like a hemispherical bowl, remember there are three separate surfaces to sum: the outer curved surface, the inner curved surface, and the flat rim — forgetting the rim is the most common mistake.

Figure 3.1 — A hemispherical bowl's three surfaces A cross-section of a hemispherical bowl with three parts labelled separately: the outer curved surface, the inner curved surface, and the flat rim connecting them. Upload image here

✈ AAI ATC relevance — direct formula-substitution mensuration questions are close to guaranteed points once the formulas are memorised cold

Memory trick — "A bowl has three surfaces, not one"

Composite solids almost always have more surfaces than they first appear to — always sketch or list every distinct surface before adding.

Moderate

Using π=22/7, the volume of a cylinder with radius 7 cm and height 10 cm is:

Volume = πr²h = (22/7)×7×7×10 = 1540 cm³.

Hard

The outer and inner radii of a hemispherical bowl are 40 cm and 35 cm respectively. Find the total surface area of the bowl.

Three surfaces: outer 2πR²=3200π, inner 2πr²=2450π, rim π(R²−r²)=375π. Total = 6025π.

Exam strategy

Facts worth having automatic

Whole vs naturalWhole numbers include 0; natural numbers don't.

Unit digit cyclesEvery power's unit digit cycles in blocks of 4.

Compound divisibilitySplit into simpler factors (e.g. 90 = 9 × 10) and test separately.

Composite solidsCount every distinct surface — bowls have three, not one.

Formula recallMensuration is substitution, not derivation — memorise the formulas cold.

Quick revision

Chapter summary sheet

Whole numbersStart at 0.

Natural numbersStart at 1.

Unit digit cycleRepeats every 4 powers.

Divisible by 90Needs both by-9 and by-10.

Rectangle areal×b.

Circle areaπr².

Cube volumea³.

Cylinder volumeπr²h.

Bowl surfacesOuter + inner + rim, three parts.

↓ Free download — Number System and Mensuration question PDF

All 5 questions with a full answer key.

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End of chapter

Comprehensive assessment

Full solutions are in the downloadable PDF above.

Multiple choice questions

1. What is the product of the smallest 8 whole numbers?

2. Find the unit digit of 7 raised to the power 45.

3. Find x+y if 3212xy is divisible by 90.

4. State the formula for the area of a trapezium.

5. Find the volume of a cylinder with radius 7 cm and height 10 cm, using π=22/7.

6. Find the total surface area of a hemispherical bowl with outer radius 40 cm and inner radius 35 cm.

7. Find the HCF of 12 and 18.

8. Find the largest three-digit number divisible by 7.

9. Find the number of factors of 36.

10. If the cost of 30 articles is Rs. 4,500, find the cost of 12 articles.