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Gyaan Maths
CBSE · Class 6-10
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CBSE Maths · Class 10

Chapter 1: Real Numbers

Name: ________________________
Date: ______________
Marks: ____ / 10

Fundamental Theorem of Arithmetic, HCF and LCM by prime factorisation, and the irrationality of numbers like 2\sqrt{2}, 3\sqrt{3}, 5\sqrt{5}.

Questions

  1. Find the HCF and LCM of 12,1512, 15 and 2121.
  2. Prove that 3+253 + 2\sqrt{5} is irrational.
  3. Can the number 6n6^n end with the digit 00?
  4. Two numbers have HCF 99 and LCM 20162016. If one is 5454, find the other.
  5. Show that any positive odd integer is of the form 4q+14q+1 or 4q+34q+3.
  6. Decide (without long division) whether 178\dfrac{17}{8} terminates.
  7. Prove that 7\sqrt{7} is irrational.
  8. Find LCM of 306306 and 657657 if HCF is 99.
  9. Find HCF of 180180 and 252252 using Euclid's algorithm.
  10. Explain why 7⋅11⋅13+137\cdot 11\cdot 13 + 13 is composite.

Extra Practice (Optional)

  • LCM of 306,657306, 657 given HCF =9=9.
  • Prove 7\sqrt 7 is irrational.
  • Show any odd integer is 4q+14q+1 or 4q+34q+3.
  • HCF of 180,252180, 252.
  • Is 29343\dfrac{29}{343} a terminating decimal?

Answer Key (For Teachers)

  1. Answer: HCF =3= 3, LCM =420= 420.
    Working: Prime factorise: 12=22⋅3,  15=3⋅5,  21=3⋅712 = 2^2\cdot 3,\; 15 = 3\cdot 5,\; 21 = 3\cdot 7. → HCF == smallest power of common primes =3= 3. → LCM == greatest power of each prime =22⋅3⋅5⋅7=420= 2^2\cdot 3\cdot 5\cdot 7 = 420.
  2. Answer: Irrational.
    Working: Assume 3+253+2\sqrt 5 is rational =p/q= p/q. → Then 5=(p−3q)/(2q)\sqrt 5 = (p-3q)/(2q) is rational. → This contradicts the fact that 5\sqrt 5 is irrational.
  3. Answer: No.
    Working: Ending in 00 needs both 22 and 55 in factorisation. → 6n=2n⋅3n6^n = 2^n\cdot 3^n has no factor 55.
  4. Answer: 336336.
    Working: Product == HCF ⋅\cdot LCM =18144= 18144. → Other =18144/54= 18144 / 54.
  5. Answer: 4q+14q+1 or 4q+34q+3.
    Working: Division lemma with b=4b=4: a=4q,4q+1,4q+2,4q+3a = 4q, 4q+1, 4q+2, 4q+3. → 4q,4q+24q, 4q+2 are even; remaining are odd.
  6. Answer: Terminates (=2.125= 2.125).
    Working: 8=23=23⋅508 = 2^3 = 2^3\cdot 5^0, of form 2m5n2^m 5^n. → Hence the decimal terminates.
  7. Answer: Irrational.
    Working: Assume 7=a/b\sqrt 7 = a/b with gcd⁡(a,b)=1\gcd(a,b)=1. → 7b2=a2⇒7∣a⇒a=7k7b^2 = a^2\Rightarrow 7\mid a\Rightarrow a=7k. → Then b2=7k2⇒7∣bb^2 = 7k^2 \Rightarrow 7\mid b; contradiction.
  8. Answer: 2233822338.
    Working: LCM =(306⋅657)/9= (306\cdot 657)/9. → =201042/9=22338= 201042/9 = 22338.
  9. Answer: HCF =36= 36.
    Working: 252=180⋅1+72252 = 180\cdot 1 + 72. → 180=72⋅2+36180 = 72\cdot 2 + 36. → 72=36⋅2+072 = 36\cdot 2 + 0.
  10. Answer: Composite.
    Working: Factor out 1313: =13(7⋅11+1)=13⋅78= 13(7\cdot 11 + 1) = 13\cdot 78. → Two natural factors >1>1.
Gyaan Maths · CBSE Class 10 · Chapter 1
Gyaan Maths — CBSE Class 6-10, built for board-toppers.
v1.1 · Feb 2026