Real Numbers · Class 10 Mathematics: 64 flashcards
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1. State the Fundamental Theorem of Arithmetic.
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Every composite number breaks into primes in one way only: the same primes with the same multiplicities, whatever order they are written in.
2. Is 1 prime, composite, or neither?
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Neither. It has only one factor.
3. What is the only even prime?
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2
4. Prime factorisation of 360 in power form?
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$2^3 \times 3^2 \times 5$
5. Prime factorisation of 4620?
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$2^2 \times 3 \times 5 \times 7 \times 11$
6. Prime factorisation of 3150?
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$2 \times 3^2 \times 5^2 \times 7$
7. What must the factorisation of a number contain for the number to end in 0?
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Both 2 and 5.
8. Can $4^n$ end in 0 for some natural number n?
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No. $4^n = 2^{2n}$ has no factor 5.
9. Can $6^n$ end in 0?
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No. $6^n = 2^n \times 3^n$ — the 5 is missing.
10. Does $20^n$ end in 0 for every natural n?
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Yes. $20^n = 2^{2n} \times 5^n$ contains both 2 and 5.
11. Show that $7 \times 11 \times 13 + 13$ is composite.
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$= 13 \times (77 + 1) = 13 \times 78$, a product of two numbers greater than 1.
12. Show that $5 \times 4 \times 3 \times 2 \times 1 + 5$ is composite.
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$= 5 \times (24 + 1) = 5 \times 25$
13. To test whether n is prime, which divisors do you need to try?
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Only the primes up to $\sqrt{n}$.
14. Is 221 prime?
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No: $221 = 13 \times 17$.
15. Is 211 prime?
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Yes — no prime up to 13 divides it.
16. What is the first step of the division ladder?
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Divide by the smallest prime, 2, as long as it divides exactly.
17. When is a factor tree finished?
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When every branch ends in a prime.
18. Is $2^3 \times 45$ the prime factorisation of 360?
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No — 45 is not prime. The factorisation is $2^3 \times 3^2 \times 5$.
19. When does m divide N, using factorisations?
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When every prime power in m appears in N with at least the same exponent.
20. $N = 2^3 \times 3^2 \times 5 \times 7$. Is N divisible by 25?
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No — N has only one factor 5, and 25 needs two.
21. Who gave the first correct proof of the Fundamental Theorem of Arithmetic?
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Carl Friedrich Gauss.
22. In a factor tree, x = 2 × a, a = 2 × b, b = 3 × c, c = 3 × 7. Find x.
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252 (c = 21, b = 63, a = 126).
23. How do you find the HCF from prime factorisations?
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Multiply the smallest power of each prime common to all the numbers.
24. How do you find the LCM from prime factorisations?
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Multiply the greatest power of every prime that appears in any of the numbers.
25. HCF and LCM of 84 and 120?
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HCF 12, LCM 840.
26. State the product rule.
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HCF(a, b) × LCM(a, b) = a × b, for two positive integers.
27. Does HCF × LCM = product hold for three numbers?
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No. For 12, 18, 30: 6 × 180 = 1080 but 12 × 18 × 30 = 6480.
28. HCF(306, 657) = 9. LCM?
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22338
29. HCF and LCM of 12, 18 and 30?
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HCF 6, LCM 180.
30. What are co-prime numbers?
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Numbers whose HCF is 1. Their LCM is their product.
31. Are two consecutive numbers always co-prime?
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Yes — a common factor would have to divide their difference, 1.
32. Which word problems use the HCF?
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Splitting into the largest equal groups: tiles, rows, pieces, boxes.
33. Which word problems use the LCM?
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Things happening together again: bells, lights, runners, buses.
34. Largest number dividing a and b leaving remainder r?
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HCF(a − r, b − r).
35. Largest number dividing 245 and 1029, leaving remainder 5 in each case?
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16, because HCF(240, 1024) = 16.
36. Can two numbers have HCF 18 and LCM 380?
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No — the HCF must divide the LCM, and 18 does not divide 380.
37. If a divides b, what are HCF(a, b) and LCM(a, b)?
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HCF = a, LCM = b.
38. Bells ring every 40, 60, 90 min, together at 8 a.m. Next together?
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2 p.m. (LCM = 360 min)
39. Largest square tile for a 420 cm × 130 cm floor?
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10 cm (HCF), using 546 tiles.
40. HCF of two numbers is 16, their product 3072. LCM?
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192
41. Where does HCF lie compared with the numbers?
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HCF ≤ the smallest number; LCM ≥ the largest.
42. LCM of $2^x \times 3^2$ and $2^3 \times 3^4 \times 5$ is $2^5 \times 3^4 \times 5$. Find x.
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x = 5
43. Another name for the HCF?
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GCD — greatest common divisor.
44. What is a rational number?
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One that can be written as $\frac{p}{q}$ with p, q integers and $q \neq 0$.
45. What is an irrational number?
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A real number that cannot be written as $\frac{p}{q}$ with p, q integers, $q \neq 0$.
46. State the theorem used in irrationality proofs.
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If a prime p divides $a^2$, then p divides a.
47. Why must p be prime in 'p | a² ⇒ p | a'?
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It fails otherwise: 4 divides 36 = 6² but not 6.
48. What are the four steps of proof by contradiction?
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Assume the opposite, reason correctly, reach an impossibility, conclude the statement is true.
49. First line of the √2 proof?
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Assume √2 = a/b, where a, b are integers, b ≠ 0, and a, b are co-prime.
50. In the √2 proof, what does squaring $\sqrt{2} = \frac{a}{b}$ give?
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$2b^2 = a^2$
51. After finding 2 | a in the √2 proof, what next?
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Write a = 2c, get $b^2 = 2c^2$, so 2 | b.
52. What is the contradiction in the √2 proof?
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2 divides both a and b, but they were assumed co-prime.
53. How do you prove 5 − √3 irrational?
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Assume it equals a/b; then $\sqrt{3} = \frac{5b - a}{b}$ is rational — contradiction.
54. Assume $3 + 2\sqrt{5} = \frac{a}{b}$. Express √5.
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$\sqrt{5} = \frac{a - 3b}{2b}$
55. Is $7\sqrt{5}$ rational or irrational?
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Irrational — a non-zero rational times an irrational.
56. Is $\frac{1}{\sqrt{2}}$ irrational?
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Yes: if it were a/b, √2 = b/a would be rational.
57. Is the sum of two irrationals always irrational?
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No: $(2 + \sqrt{3}) + (2 - \sqrt{3}) = 4$.
58. Is the product of two irrationals always irrational?
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No: $\sqrt{2} \times \sqrt{2} = 2$.
59. $(3 + \sqrt{5})(3 - \sqrt{5}) = ?$
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4 — rational.
60. $(\sqrt{2} + \sqrt{3})^2 = ?$
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$5 + 2\sqrt{6}$ — irrational.
61. Is $\sqrt{\frac{4}{9}}$ irrational?
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No, it is $\frac{2}{3}$.
62. When is $\sqrt{n}$ irrational (n a natural number)?
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When n is not a perfect square.
63. Why does 'rational × irrational = irrational' need a non-zero rational?
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Because 0 × √2 = 0 is rational.
64. What does 'reductio ad absurdum' mean?
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Proof by contradiction — reduce the assumption to an absurdity.
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