Real Numbers — Practice

Question 1

Find the HCF of 867 and 255 using Euclid's Division Algorithm.

Accepted Answer

Step 1 — Apply the lemma to 867 and 255 (larger first).
  867 = 255 × 3 + 102

Step 2 — Apply again to 255 and 102.
  255 = 102 × 2 + 51

Step 3 — Apply to 102 and 51.
  102 = 51 × 2 + 0

Remainder is 0. Last non-zero remainder = 51.

✦ Answer: HCF(867, 255) = 51.

Question 2

Find the HCF and LCM of 336 and 54 using prime factorisation. Verify HCF × LCM = product of the two numbers.

Accepted Answer

Step 1 — Prime factorise each number.
  336 = 2⁴ × 3 × 7
  54 = 2 × 3³

Step 2 — Find HCF (smallest powers of COMMON factors).
  Common primes: 2 and 3.
  HCF = 2¹ × 3¹ = 6.

Step 3 — Find LCM (largest powers of ALL factors).
  LCM = 2⁴ × 3³ × 7 = 16 × 27 × 7 = 3024.

Step 4 — Verify.
  HCF × LCM = 6 × 3024 = 18144.
  Product of numbers = 336 × 54 = 18144. ✓

✦ Answer: HCF = 6, LCM = 3024.

Question 3

Without performing actual division, classify whether 29/343 is a terminating or non-terminating recurring decimal.

Accepted Answer

Step 1 — Factorise the denominator.
  343 = 7³.

Step 2 — Check prime factors.
  Denominator has factor 7 (which is NOT 2 or 5).

Conclusion: The decimal expansion is NON-TERMINATING RECURRING.

✦ Answer: 29/343 is non-terminating recurring.

Question 4

Prove that √3 is irrational.

Accepted Answer

Step 1 — Assume, for contradiction, that √3 is rational.
  So √3 = p/q where p, q are integers, q ≠ 0, and HCF(p, q) = 1 (in lowest terms).

Step 2 — Square both sides.
  3 = p²/q² → 3q² = p².

Step 3 — Deduce p is divisible by 3.
  p² is divisible by 3. By the fundamental lemma (if prime p divides a², then p divides a), p is divisible by 3.
  So let p = 3k for some integer k.

Step 4 — Substitute back.
  3q² = (3k)² = 9k² → q² = 3k².

Step 5 — Deduce q is divisible by 3.
  q² is divisible by 3, so q is divisible by 3.

Step 6 — Contradiction.
  Both p and q are divisible by 3 → HCF(p, q) ≥ 3.
  But we assumed HCF(p, q) = 1. CONTRADICTION.

Conclusion: Our assumption that √3 is rational is FALSE.

✦ Answer: Therefore, √3 is irrational. ∎

Question 5

Prove that 3 + 2√5 is irrational.

Accepted Answer

Step 1 — Assume 3 + 2√5 is rational.
  Then 3 + 2√5 = p/q where p, q are integers, q ≠ 0.

Step 2 — Isolate √5.
  2√5 = p/q − 3 = (p − 3q)/q.
  √5 = (p − 3q)/(2q).

Step 3 — Analyse the right side.
  Since p, q are integers: (p − 3q) and (2q) are also integers, with 2q ≠ 0.
  So the right side is a RATIONAL number.

Step 4 — Contradiction.
  This implies √5 is rational. But √5 is known to be IRRATIONAL.
  CONTRADICTION.

Conclusion: Our assumption is FALSE.

✦ Answer: Therefore, 3 + 2√5 is irrational. ∎

Question 6

Three pieces of ribbon have lengths 98 cm, 140 cm, and 168 cm. What is the maximum length of ribbon pieces that can be cut from all three ribbons without any wastage?

Accepted Answer

Step 1 — We need the HCF of 98, 140, and 168.
  98 = 2 × 7²
  140 = 2² × 5 × 7
  168 = 2³ × 3 × 7

Step 2 — HCF = product of SMALLEST powers of COMMON factors.
  Common factors: 2 and 7.
  HCF = 2¹ × 7¹ = 14.

✦ Answer: Maximum length of each cut piece = 14 cm.

Verification: 98 ÷ 14 = 7, 140 ÷ 14 = 10, 168 ÷ 14 = 12 — all divide exactly. ✓

Question 7

Prove that the square of any positive integer is of the form 5q, 5q+1, or 5q+4.

Accepted Answer

Step 1 — Every positive integer can be written as 5m, 5m+1, 5m+2, 5m+3, or 5m+4 (Euclid's Lemma with b=5).

Step 2 — Square each form.
  Case 1: n = 5m → n² = 25m² = 5(5m²) = 5q ✓
  Case 2: n = 5m+1 → n² = 25m²+10m+1 = 5(5m²+2m)+1 = 5q+1 ✓
  Case 3: n = 5m+2 → n² = 25m²+20m+4 = 5(5m²+4m)+4 = 5q+4 ✓
  Case 4: n = 5m+3 → n² = 25m²+30m+9 = 5(5m²+6m+1)+4 = 5q+4 ✓
  Case 5: n = 5m+4 → n² = 25m²+40m+16 = 5(5m²+8m+3)+1 = 5q+1 ✓

Step 3 — Conclusion.
  In all cases, n² is of the form 5q, 5q+1, or 5q+4 (never 5q+2 or 5q+3).

✦ Answer: Proved. ∎

Practice — Real Numbers

Sign in to save attempts to your dashboard and track your accuracy over time

Find a verified tutor

Join a live course

Browse coaching centres

About

Categories

Support

Documentation

Get Study Tips & Updates

Download Our App

Download for iOS

Download for Android

Popular Cities