Number System & Simplification

Natural numbers (N): 1,2,3,... Whole numbers (W): 0,1,2,... Integers (Z): ...-2,-1,0,1,2... Rational numbers (Q): expressible as p/q, q not 0 (includes terminating and recurring decimals). Irrational numbers: non-terminating, non-recurring (root2, pi, e). Real numbers = rational + irrational. Prime: exactly two factors (1 and itself); 2 is the only even prime. Composite: more than two factors. Co-prime: HCF = 1 (e.g., 8 and 15). Memory aid: '0 is whole but not natural; 1 is neither prime nor composite.' Even numbers end in 0,2,4,6,8; odd end in 1,3,5,7,9. There are 25 prime numbers below 100 — remember this count for GA-style number questions.

By 2: last digit even. By 3: digit-sum divisible by 3. By 4: last two digits divisible by 4. By 5: ends in 0 or 5. By 6: divisible by both 2 and 3. By 8: last three digits divisible by 8. By 9: digit-sum divisible by 9. By 10: ends in 0. By 11: (sum of odd-place digits) - (sum of even-place digits) = 0 or multiple of 11. By 25: last two digits 00,25,50,75. Shortcut for 7: double the last digit, subtract from the rest; if result divisible by 7, the number is too (e.g., 203: 20 - 6 = 14, divisible by 7).

Check if 4,832,718 is divisible by 11. Mark places from the right: digits 8(1),1(2),7(3),2(4),3(5),8(6),4(7). Odd-place (1,3,5,7) digits: 8+7+3+4 = 22. Even-place (2,4,6) digits: 1+2+8 = 11. Difference = 22 - 11 = 11, which is a multiple of 11, so the number IS divisible by 11. This alternating-sum method is far faster than long division and is the most-tested divisibility rule in RPF SI papers, so practice tagging odd/even positions from the rightmost digit.

HCF (Highest Common Factor) = the largest number dividing all given numbers. LCM (Lowest Common Multiple) = smallest number divisible by all. KEY RELATION: for any two numbers a and b, HCF x LCM = a x b. This lets you find one when three values are known. HCF is always less than or equal to the smallest number; LCM is always greater than or equal to the largest number. HCF always divides LCM exactly. For fractions: HCF of fractions = HCF(numerators)/LCM(denominators); LCM of fractions = LCM(numerators)/HCF(denominators). Memory aid: 'Highest Cuts Fractions' (HCF uses HCF on top), opposite for LCM.

Prime factorisation method: write each number as a product of primes. HCF = product of the LOWEST powers of common primes. LCM = product of the HIGHEST powers of all primes appearing. Example: 12 = 2^2 x 3, 18 = 2 x 3^2. HCF = 2^1 x 3^1 = 6; LCM = 2^2 x 3^2 = 36. Check: 6 x 36 = 216 = 12 x 18. The division (ladder) method is faster for LCM of many numbers: divide repeatedly by common primes, then multiply all divisors and remaining quotients. For HCF of large numbers, the long-division (Euclidean) method is quickest.

BELLS RINGING TOGETHER: if bells toll at intervals of 6, 9, 12 seconds, they ring together after LCM(6,9,12) = 36 seconds; in 1 hour (3600 s) together = 3600/36 + 1 times (the +1 counts the start). COMMON REMAINDER: 'smallest number which when divided by a, b, c leaves the same remainder r' = LCM(a,b,c) + r. 'Largest number dividing x, y, z leaving remainder r each' = HCF(x-r, y-r, z-r), or HCF of the differences when remainders are equal but unknown. These two templates cover the majority of RPF SI HCF/LCM word problems — identify which one the question fits first.

BODMAS fixes the sequence for any simplification: B - Brackets (solve innermost first: (), {}, []), O - Of / Order (powers, roots, 'of' means multiply), D - Division, M - Multiplication, A - Addition, S - Subtraction. Division and multiplication rank equally — work left to right; same for addition and subtraction. The word 'of' is treated as multiplication but is performed before normal division (e.g., 1/2 of 8 / 4 = 4/4 = 1). Memory aid: 'Brackets Order DM AS.' A single misstep in operation order is the most common reason candidates lose marks in the simplification section — always rewrite the expression bracket-by-bracket.

Some boards use VBODMAS where V = Vinculum (bar/line), solved first — e.g., in 12 - (8 - bar over 3-1), simplify the bar 3-1=2 first. For mixed fractions, convert to improper fractions before applying BODMAS. Cancel common factors early to avoid large numbers. For chained 'of' and division: 'of' binds tighter, so 24 / 4 of 3 = 24 / 12 = 2 (do the 'of' multiplication first). When you see surds or squares, evaluate Order before Division. Shortcut: replace repeated decimals like 0.333... with fractions (1/3) to keep arithmetic exact rather than approximate.

Simplify: 36 - [18 - {14 - (15 - 6 / 3 x 2)}]. Step 1 innermost: 6/3 = 2, then 2 x 2 = 4, so (15 - 4) = 11. Step 2 next bracket: {14 - 11} = 3. Step 3: [18 - 3] = 15. Step 4: 36 - 15 = 21. Answer = 21. Note how division and multiplication inside the parentheses were done left to right BEFORE the subtraction. Working strictly from the innermost bracket outward prevents sign errors — a frequent trap in RPF SI multi-bracket simplification problems.

Master these eight laws: (1) a^m x a^n = a^(m+n); (2) a^m / a^n = a^(m-n); (3) (a^m)^n = a^(mn); (4) (ab)^n = a^n x b^n; (5) (a/b)^n = a^n / b^n; (6) a^0 = 1 (a not 0); (7) a^(-n) = 1/a^n; (8) a^(m/n) = nth root of a^m. To compare powers, make either the base or the exponent equal. Memory aid: 'Same base ADD/SUBTRACT, power-of-power MULTIPLY.' These laws drive nearly all RPF SI surd-and-indices questions; converting roots to fractional powers (root a = a^(1/2)) often turns a hard surd problem into simple index arithmetic.

A surd is an irrational root like root 2 or cube root 5. Rules: root a x root b = root(ab); root a / root b = root(a/b). RATIONALISING removes surds from the denominator: multiply numerator and denominator by the conjugate. For 1/(root a) multiply by root a. For 1/(a + root b), multiply by the conjugate (a - root b) so the denominator becomes a^2 - b (difference of squares). Example: 1/(3 + root 2) x (3 - root 2)/(3 - root 2) = (3 - root 2)/(9 - 2) = (3 - root 2)/7. Always simplify surds to lowest form: root 50 = root(25 x 2) = 5 root 2.

Perfect-square unit digits can only be 0,1,4,5,6,9 — a number ending in 2,3,7,8 is never a perfect square. To find a 4-digit square root, split into pairs from the right; the tens digit comes from the larger pair, the units from the ending digit (5 is the giveaway since only 25 ends in 5). For cube roots of perfect cubes, the unit digit of the cube reveals the root's unit digit (cubes end: 1→1, 8→2, 7→3, 4→4, 5→5, 6→6, 3→7, 2→8, 9→9, 0→0). Example: cube root of 19683 — ends in 3 so unit is 7, and 27 (=3^3) < 19 < 64 so tens is 2, giving 27.