Number System & Simplification

Natural numbers (N): 1,2,3,... Whole numbers (W): 0,1,2,3,... (add zero to N). Integers (Z): ...-2,-1,0,1,2... Rational numbers: can be written as p/q (q not 0), e.g. 3/4, 0.25, 0.333. Irrational numbers: non-terminating non-repeating, e.g. root2, pi. Prime number: exactly two factors (1 and itself); 2 is the smallest and ONLY even prime. Composite: more than two factors. 1 is NEITHER prime nor composite. Even: divisible by 2; Odd: not. Memory aid: every Natural is Whole, every Whole is Integer, every Integer is Rational. There are 25 prime numbers below 100. Co-prime numbers share only 1 as common factor, e.g. (8,15).

By 2: last digit even (0,2,4,6,8). By 3: digit-sum divisible by 3. By 4: last two digits divisible by 4. By 5: last digit 0 or 5. By 6: divisible by 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-position digits) minus (sum of even-position digits) is 0 or a multiple of 11. By 7 shortcut: double the last digit, subtract from rest; if result divisible by 7 the number is. Trick for 6,12,15: check the two co-prime factors (e.g. 12 = 3 and 4). Always test smaller co-prime factors rather than the big number itself.

Check if 4,83,71 (483771) is divisible by 11. Mark positions from the right: digit positions are 1(odd),7(even),7(odd),3(even),8(odd),4(even). Odd-position digits: 1+7+8 = 16. Even-position digits: 7+3+4 = 14. Difference = 16 - 14 = 2. Since 2 is NOT 0 or a multiple of 11, 483771 is NOT divisible by 11. Compare with 4,83,769: odd positions 9+7+8=24, even 6+3+4=13, difference 11 which IS a multiple of 11, so 483769 is divisible by 11. Always keep position counting consistent (start from the right or left, just be uniform).

HCF (Highest Common Factor) = the largest number that divides all given numbers exactly. LCM (Least Common Multiple) = the smallest number divisible by all given numbers. Methods: Prime factorisation - HCF = product of common prime factors with LOWEST powers; LCM = product of all prime factors with HIGHEST powers. Division method is faster for two large numbers. Golden formula: HCF x LCM = Product of the two numbers (works for exactly two numbers). Memory aid: HCF is small (Highest common but a divisor, so it is the smaller value), LCM is large (a multiple, so it is the bigger value). HCF of co-prime numbers is always 1, and their LCM equals their product.

For fractions, use these special formulas. HCF of fractions = HCF of numerators / LCM of denominators. LCM of fractions = LCM of numerators / HCF of denominators. Example: HCF of 2/3 and 4/9 = HCF(2,4)/LCM(3,9) = 2/9. LCM of 2/3 and 4/9 = LCM(2,4)/HCF(3,9) = 4/3. Memory hook: for HCF take the small route (HCF on top, LCM on bottom); for LCM take the big route (LCM on top, HCF on bottom). Always reduce fractions to lowest terms first. These appear regularly in RPF and other RRB Level-1 exams as one-mark direct questions.

The HCF of two numbers is 12 and their LCM is 144. If one number is 36, find the other. Use the rule: Product of numbers = HCF x LCM. So 36 x other = 12 x 144 = 1728. Therefore other = 1728 / 36 = 48. Verify: HCF(36,48) = 12, LCM(36,48) = 144. Correct. This product rule ONLY works for two numbers. A common bell-ringing or traffic-light LCM word problem: lights blink at 24, 36 and 54 seconds; they blink together after LCM(24,36,54) = 216 seconds = 3 min 36 sec.

BODMAS gives the order in which operations are solved: B - Brackets, O - Of (powers/orders), D - Division, M - Multiplication, A - Addition, S - Subtraction. Division and Multiplication rank EQUAL - do them left to right. Same for Addition and Subtraction. Bracket order: solve innermost first - ( ) then { } then [ ]. The word 'of' means multiply but is done before division (e.g. 1/2 of 8 = 4 first). Memory aid: 'Brackets Open, Divide Multiply, Add Subtract'. A frequent trap: students do addition before multiplication - always finish multiply/divide first. Sign rule: +x+ = +, -x- = +, +x- = -, -x+ = -.

'Of' is treated as multiplication but evaluated immediately after brackets and before plain division. Example: 1/3 of 12 / 2 = (1/3 x 12) / 2 = 4 / 2 = 2 - do the 'of' first. When simplifying fractions, convert mixed numbers to improper fractions, then apply BODMAS. To divide by a fraction, multiply by its reciprocal: a / (b/c) = a x c/b. For a 'fraction bar' that acts as a bracket, simplify the numerator and denominator separately before dividing. Memory tip: vinculum (bar over numbers like a line) is solved FIRST, even before brackets. Keep signs careful when removing brackets preceded by a minus sign - every inside term flips sign.

Simplify: 36 / 4 x 3 + 8 - 5 x 2. Step 1 (Division/Multiplication, left to right): 36/4 = 9, then 9 x 3 = 27; also 5 x 2 = 10. Expression becomes 27 + 8 - 10. Step 2 (Addition/Subtraction, left to right): 27 + 8 = 35, then 35 - 10 = 25. Answer = 25. Common mistake: doing 4 x 3 = 12 first then 36/12 = 3 - WRONG, because division comes before this multiplication when reading left to right. Always process division and multiplication strictly from left to right, never by which symbol looks first.

Proper fraction: numerator < denominator (3/5). Improper: numerator >= denominator (7/4). Mixed number: whole + fraction (1 3/4). To convert fraction to decimal, divide numerator by denominator. To convert a terminating decimal to a fraction, write digits over the place value (0.75 = 75/100 = 3/4). For comparing fractions, cross-multiply or make denominators equal. To add/subtract fractions, take LCM of denominators. To multiply, multiply across; to divide, multiply by the reciprocal. Memory aid for recurring decimals: 0.333... = 1/3, 0.666... = 2/3, 0.142857... = 1/7. Ascending order trick: smaller decimal value = smaller fraction. Always reduce final answers to lowest terms.

A surd is an irrational root like root2, root3, root5. Rules: root(a) x root(b) = root(ab); root(a)/root(b) = root(a/b); root(a^2 x b) = a root(b). Useful values to memorise: root2 = 1.414, root3 = 1.732, root5 = 2.236, root7 = 2.646. Rationalising the denominator: multiply numerator and denominator by the surd, e.g. 1/root2 = root2/2. For square roots of perfect squares, learn squares up to 30 (e.g. 25^2 = 625, 30^2 = 900). To find square root by factorisation, pair the prime factors: root324 = root(2^2 x 3^4) = 2 x 9 = 18. These shortcuts save crucial seconds in the exam.

Simplify 2/3 + 3/4 - 1/6. Step 1: LCM of 3, 4 and 6 is 12. Convert: 2/3 = 8/12, 3/4 = 9/12, 1/6 = 2/12. Step 2: 8/12 + 9/12 - 2/12 = (8 + 9 - 2)/12 = 15/12. Step 3: reduce 15/12 = 5/4 = 1 1/4. Answer = 5/4. For arranging in ascending order, e.g. 2/3, 3/5, 4/7: convert to decimals - 0.667, 0.600, 0.571 - so ascending order is 4/7 < 3/5 < 2/3. Decimal conversion is the fastest, error-free way to compare unlike fractions under exam time pressure.