All natural numbers together with 0 form which set of numbers?

Whole numbers. Natural numbers are 1, 2, 3, and so on. Adding 0 to this set gives 0, 1, 2, 3, .... This exact set -- all natural numbers plus zero -- is by definition called the whole numbers. So the answer is whole numbers.

Which of these numbers is irrational?

√7. A number is irrational if it cannot be written as a ratio of two integers, i.e. its decimal never terminates and never repeats. Check each option: 22/7 is a ratio of integers, so it's rational. 0.375 terminates, so it's rational. 1.5 terminates, so it's rational. √7 = 2.6457513... never terminates or repeats, since 7 is not a perfect square. So √7 is the irrational one.

If m and n are both odd numbers, which of these expressions is always even?

m + n. Let m = 2a+1 and n = 2b+1 for some integers a, b (the general form of an odd number). Then m+n = 2a+1+2b+1 = 2a+2b+2 = 2(a+b+1), which is divisible by 2, so m+n is always even. Check with an example: m=3, n=5, m+n=8, which is even. So the answer is m + n.

A number is divisible by 3 exactly when which condition holds?

The sum of all its digits is divisible by 3. This is the digit-sum rule: add up every digit of the number, and check if that sum is divisible by 3. For example, 123 has digit sum 1+2+3=6, which is divisible by 3, and indeed 123=3x41. This same digit-sum rule also works for divisibility by 9, but not for most other divisors like 4, 7, or 8.

How many prime factors, counted with multiplicity, are in the expression 5^9 x 7^9 x 3^8?

26. The expression is 5^9 x 7^9 x 3^8. Counting prime factors with multiplicity means counting each prime as many times as its exponent says: 5 appears 9 times, 7 appears 9 times, 3 appears 8 times. Total count = 9 + 9 + 8 = 26.

In how many distinct ways can 11 be expressed as the sum of two different positive integers?

5. We need pairs (a, b) with a+b=11, a≠b, both positive, counting each pair once. Pairing 1 with 10, 2 with 9, and so on: each pair (a, 11-a) is valid as long as a < 11-a, i.e. a < 11/2 = 5.5. So a ranges from 1 to 5, giving 5 distinct unordered pairs. Check: the last pair is (5, 6), and 5 ≠ 6 confirms they're different numbers.

If a + b + c = 7 and ab + bc + ca = 6, what is the value of a³ + b³ + c³ - 3abc?

217. Let S1 = a+b+c = 7 and S2 = ab+bc+ca = 6. We need a³+b³+c³-3abc, which factors as (a+b+c)(a²+b²+c²-ab-bc-ca). First find a²+b²+c² using (a+b+c)² = a²+b²+c²+2(ab+bc+ca): a²+b²+c² = S1² - 2xS2 = 7² - 2x6 = 49 - 12 = 37. Then a²+b²+c²-ab-bc-ca = 37 - 6 = 31. Finally, a³+b³+c³-3abc = S1 x 31 = 7 x 31 = 217.

P and Q are positive integers such that PQ = 84. Which of these values can NOT be P + Q?

90. Since PQ = 84, P and Q must be a factor pair of 84. Listing the factor pairs of 84 and their sums: for each divisor d of 84, P=d and Q=84/d give a sum of d+84/d. This produces the possible sums 19, 20, 25, 31, 44, 85. The value 90 is not among these possible sums, so it can NOT be P+Q.

Number System: Formulas, Shortcuts and Solved Examples

The number system is the map of every kind of number you will meet.

This lesson gives each rule with the reason it works. It solves five examples step by step and lists the traps that cost marks.

The idea in brief

Order: natural, whole, integer, rational, real.
Irrational: √7 and π.
Divisibility: 3 and 9 use the digit sum.
Prime factors: add the powers.
Cube identity: (a + b + c)(a² + b² + c² − ab − bc − ca).

What the Number System Covers

What are the types of numbers?

Natural numbers are 1, 2, 3 and so on. Adding 0 gives the whole numbers.

Adding the negatives gives the integers.

A number that can be written as p/q is rational. A number whose decimal never ends or repeats is irrational.

Rational and irrational numbers together make the real numbers.

Each set sits inside the next. Natural numbers sit inside whole numbers, then integers, then rational numbers, then real numbers. Irrational numbers fill the rest of the reals.

How do I classify a number quickly?

√7 is irrational, because 7 is not a perfect square.

π is irrational. Its decimal runs on without a repeating pattern.

Every rational number is also a real number.

Not every real number is rational. That is the usual trap.

What are the odd and even rules?

Odd + odd = even. Even + even = even. Odd + even = odd.

Odd × odd = odd. Any number × even = even.

Why it works: an odd number is 2a + 1. Two of them add to 2a + 2b + 2, which is a multiple of 2.

Check 1

Every real number is a rational number.

Core Rules and Why They Work

What are the divisibility rules?

By 3: the digit sum is divisible by 3.
By 4: the last two digits are divisible by 4.
By 6: the number is divisible by both 2 and 3.
By 8: the last three digits are divisible by 8.
By 11: odd-place digit sum minus even-place digit sum is 0 or a multiple of 11.

Why 4 and 8 work: 100 is a multiple of 4 and 1000 is a multiple of 8. So only the last digits matter.

How do I write a number using its digits?

A 3-digit number with digits p, q, r is 100p + 10q + r.

A 4-digit number with digits w, x, y, z is 1000w + 100x + 10y + z.

Why it works: each place is worth ten times the place to its right.

How do I count prime factors?

Add the powers of the primes. In 5⁹ × 7⁹ × 3⁸ the count is 9 + 9 + 8 = 26.

Why it works: each power means that prime is multiplied that many times.

What is the a³ + b³ + c³ − 3abc identity?

a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca).

To use it, find a² + b² + c² from (a + b + c)² − 2(ab + bc + ca).

Why it works: expanding the right side cancels every mixed term and leaves the cubes.

What is the difference of squares?

a² − b² = (a + b)(a − b).

So (a² − b²) ÷ (a + b) is just a − b.

Why it works: multiply out (a + b)(a − b) and the middle terms cancel.

The rectangles with area 12 and their sums. When P × Q = 12, the sum P + Q can only be 13, 8 or 7.

What is the zero-product rule?

If ab = 0, then a = 0, or b = 0, or both.

Why it works: two non-zero numbers can never multiply to zero.

Check 2

A number is divisible by 6 only if it is divisible by both 2 and 3.

Five Solved Examples

Easy, step by step: how many prime factors, counted with multiplicity, does 5⁹ × 7⁹ × 3⁸ have?

Step 1: each power says how many times that prime appears.

Step 2: add the powers. 9 + 9 + 8 = 26.

The answer is 26.

Easy, step by step: in how many ways can 11 be written as the sum of two different positive integers?

Step 1: list the pairs. 1 + 10, 2 + 9, 3 + 8, 4 + 7, 5 + 6.

Step 2: count each pair once. The reverse order is the same pair.

The answer is 5.

Shortcut: for an odd number n, the count is (n − 1) ÷ 2.

Medium, step by step: a + b + c = 7 and ab + bc + ca = 6. Find a³ + b³ + c³ − 3abc.

Step 1: square the sum. 7² = 49, so a² + b² + c² = 49 − 2 × 6 = 37.

Step 2: the bracket is 37 − 6 = 31.

Step 3: multiply by the sum. 7 × 31 = 217.

The answer is 217.

Medium, step by step: P and Q are positive integers with PQ = 84. Can P + Q be 90?

Step 1: list the factor pairs of 84. They are 1 × 84, 2 × 42, 3 × 28, 4 × 21, 6 × 14 and 7 × 12.

Step 2: add each pair. The sums are 85, 44, 31, 25, 20 and 19.

Step 3: 90 is not in the list.

So P + Q cannot be 90.

Exam level, step by step: is 2728 divisible by 8 and by 11?

Step 1: for 8, look at the last three digits, 728. 728 ÷ 8 = 91, so yes.

Step 2: for 11, odd-place digits are 2 and 2, which add to 4. Even-place digits are 7 and 8, which add to 15.

Step 3: 15 − 4 = 11, a multiple of 11, so yes.

Both tests pass. In fact 2728 = 8 × 11 × 31.

Check 3

If P × Q = 84, then P + Q can be 90.

Common Traps and Shortcuts

Which traps cost marks?

Calling every real number rational.

Counting 1 + 10 and 10 + 1 as two different ways.

Using the digit-sum rule for 4, 8 or 11. It works only for 3 and 9.

Forgetting that 0 is a whole number but not a natural number.

Forgetting the sign when an even power is involved.

Which shortcuts save time?

Test numbers like 3 and 4 for odd/even questions. A single counter-example kills a wrong option.

For the 11 rule, alternate plus and minus across the digits.

For “cannot be P + Q” questions, list the factor pairs first.

Use the identity before expanding any bracket.

What patterns come up in exams?

Six types repeat. Classify a number, count prime factors, count pairs, use a cube identity, test divisibility and judge odd or even.

Each can be done in under a minute if you know the rule.

Check 4

The digit-sum test works for divisibility by 3 and 9.

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