Quick Revision

Number Sets and the Box Picture

$$\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$$

• Natural: 1, 2, 3, … (no 0)
• Whole: 0, 1, 2, 3, …
• Integers: whole numbers and their negatives
• Rational: $\frac{p}{q}$ with $q \neq 0$, including terminating and recurring decimals
• Irrational: never ends and never repeats, like $\sqrt{2}$ and $\pi$
• Real: rational and irrational together

Traps: 0 is whole but not natural. "Integer but not whole" means negative. $\sqrt{16} = 4$ is rational. 22/7 is rational, $\pi$ is not. No number is both rational and irrational.

Rational and Irrational Operations

• Rational ± rational, rational × rational: always rational
• Rational ± irrational: always irrational
• Non-zero rational × irrational: always irrational
• Irrational ± irrational, irrational × irrational: can be either
• $\sqrt{n}$ is irrational unless $n$ is a perfect square

Examples: $\sqrt{2} \times \sqrt{8} = 4$ is rational, $\sqrt{2} \times \sqrt{3} = \sqrt{6}$ is irrational, and $(5 + \sqrt{3}) + (5 - \sqrt{3}) = 10$ is rational.

[SSC Tip] Simplify before you decide. $(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$ is always rational when $a$ and $b$ are rational. The rule says non-zero because $0 \times \sqrt{2} = 0$.

Terminating and Recurring Decimals

• Reduce the fraction to lowest terms first.
• It terminates if the denominator has only 2s and 5s, and recurs otherwise. 21/56 = 3/8 terminates even though 56 has a 7.
• For $\frac{a}{2^m \times 5^n}$, decimal places = the larger of $m$ and $n$.

Converting:

• Pure: $0.\overline{45} = \frac{45}{99}$, one 9 per repeating digit
• Mixed: $0.1\overline{6} = \frac{16 - 1}{90}$, a 9 per repeating digit, then a 0 per non-repeating digit
• $0.\overline{9} = 1$

Add recurring decimals as fractions, never digit by digit.

Primes, Co-primes and Digit Facts

• 1 is neither prime nor composite. 2 is the only even prime.
• There are 25 primes up to 100 (15 up to 50).
• To test $n$, divide by the primes up to $\sqrt{n}$.
• Co-prime means HCF 1. Consecutive numbers are always co-prime, and 8 and 15 are co-prime though neither is prime.
• Lookalikes that are not prime: 91, 119, 133, 143, 161, 187, 203, 209, 221.
• Place value = digit × value of its place. Face value = the digit.
• Number − reverse = $9(a - b)$. Number + reverse = $11(a + b)$.

Divisibility Rules

1001 = 7 × 11 × 13, so any six-digit number like abcabc is divisible by all three.

Composite Divisors and Missing Digits

• Split into co-prime factors: 24 = 3 × 8, 36 = 4 × 9, 72 = 8 × 9, 88 = 8 × 11, 99 = 9 × 11. Never 4 × 6 for 24 (they share a 2).
• Missing digits: use the last-digits rule (5, 4, 8) first, then the digit-sum or 11-test.
• For 99, add digit pairs from the right; the total must be a multiple of 99.
• Least to subtract = remainder. Least to add = divisor − remainder.
• Largest n-digit multiple = largest n-digit number − remainder.

Counting and Summing Multiples

• Multiples of $k$ from 1 to $N$: the whole-number part of $N \div k$.
• Divisible by $a$ and $b$ together: count multiples of their LCM.
• Divisible by $a$ but not $b$: multiples of $a$ minus multiples of LCM($a$, $b$).
• $1 + 2 + \dots + n = \frac{n(n + 1)}{2}$. The first $n$ odd numbers add to $n^2$.
• Sum of multiples in a range = count × (first + last) ÷ 2.
• $a^n - b^n$ is divisible by $a - b$ always, and by $a + b$ when $n$ is even.
• $a^n + b^n$ is divisible by $a + b$ when $n$ is odd.

Factor Formulas

For $N = p^a \times q^b \times r^c$:

• Number of factors = $(a + 1)(b + 1)(c + 1)$
• Odd factors: leave out the 2s and use the same formula on the rest
• Even factors = total − odd
• Square factors: count only even powers. $2^4$ gives 0, 2, 4, which is 3 choices
• [Tier 2] Sum of factors = $(1 + p + \dots + p^a)(1 + q + \dots + q^b)(1 + r + \dots + r^c)$

Example: 540 = $2^2 \times 3^3 \times 5$ has 24 factors, 8 odd and 16 even. A perfect square is the only kind of number with an odd number of factors.

Trailing Zeros and Highest Power

• Zeros at the end of $n!$: add $n \div 5$, $n \div 25$, $n \div 125$, … and ignore every remainder. $100!$ ends in 20 + 4 = 24 zeros.
• Highest power of a prime $p$ in $n!$: same method with $p$, $p^2$, $p^3$, …
• [Tier 2] For a composite like $12 = 2^2 \times 3$, find the powers of 2 and of 3 separately, halve the count of 2s, and take the smaller answer.
• Zeros at the end of a product: count the 2s and 5s; zeros = the smaller count.

Perfect Squares

• A square ends in 0, 1, 4, 5, 6 or 9, never in 2, 3, 7 or 8.
• Zeros at the end come in pairs: 1600 can be a square, 16000 can't.
• Every prime in its factorisation has an even power.
• Smallest multiplier or divisor: the product of the primes with odd powers.
• Least to add or subtract: find the squares just above and just below.
• Squares between A and B: count the numbers whose squares lie in the range.
• Least square divisible by given numbers: take the LCM, then make every power even.

Remainders and Changing the Divisor

• Dividend = divisor × quotient + remainder, with remainder smaller than divisor.
• Sums and products: work with the remainders, then reduce again.
• Changing the divisor: if N leaves r with D and d divides D, then N leaves the remainder of r ÷ d. Otherwise the answer is not fixed.
• Negative remainders save time: 17 leaves −1 with 18. Add the divisor back at the end.
• Same remainder r with every divisor: N = LCM × k + r.
• Same gap c = divisor − remainder: N = LCM × k − c.

Large Powers and Unit Digit Cycles

• For a big power, find a small power that leaves 1 or −1: $2^4 = 16$ leaves 1 with 5 and −1 with 17.
• $(D - 1)^n$ leaves 1 with D when $n$ is even, and D − 1 when $n$ is odd.
• [Tier 2] For a prime $p$ not dividing $a$, $a^{p-1}$ leaves 1 with $p$.
• Unit digits: 0, 1, 5, 6 never change. 4 and 9 repeat every 2 powers. 2, 3, 7, 8 repeat every 4: 2-4-8-6, 3-9-7-1, 7-9-3-1, 8-4-2-6.
• Divide the power by 4; remainder 0 means the 4th digit.

HCF and LCM Rules

• HCF: common primes, lowest powers. LCM: all primes, highest powers.
• HCF × LCM = product, for two numbers only.
• HCF always divides LCM. Co-prime: HCF 1, LCM = product.
• Ratio $a : b$ (co-prime) with HCF $h$: numbers $ah$ and $bh$, LCM $abh$.
• Fractions: HCF = HCF of numerators ÷ LCM of denominators. LCM = LCM of numerators ÷ HCF of denominators.
• Decimals: equalise the decimal places, work with whole numbers, then put the point back.
• Pairs with HCF $h$ and product P: count co-prime pairs with product $P \div h^2$.

HCF and LCM Word-Problem Signals

HCF words: greatest, largest, maximum, equal pieces, largest tile, longest rope, biggest container, most students to share equally.

LCM words: least, smallest, ring or flash together again, meet at the start, least number arranged in rows of …, smallest square from tiles.

• Same remainder r: greatest divisor = HCF of (each number − r); least number = LCM + r.
• Different remainders: subtract each remainder first, then take the HCF.
• Change to one unit first (m to cm, minutes to seconds).
• Check whether "how many times together" counts the start.