Class 10 / Maths

Class 10 Maths Real Numbers Worksheet

Practice HCF, LCM, Euclid division algorithm, prime factorization, and irrational number reasoning.

advanced45 minutesAnswer key included

Learning objective

Practise Euclid's algorithm, HCF, LCM, composite-number reasoning, and irrationality proofs.

Questions

  1. Use Euclid's division algorithm to find the HCF of 252 and 105.
  2. Find the HCF and LCM of 96 and 404 by prime factorization.
  3. Verify HCF(a, b) x LCM(a, b) = a x b for a = 45 and b = 75.
  4. Find the largest number that divides 1251, 9377, and 15628 leaving the same remainder in each case.
  5. Show that 3 x 5 x 7 + 7 is a composite number.
  6. Explain why 7 x 11 x 13 + 13 is not a prime number.
  7. Prove that sqrt(3) is irrational.
  8. Show that 2 + sqrt(5) is irrational.
  9. Find the smallest number which is divisible by 12, 15, 20, and 27.
  10. Two bells ring at intervals of 12 minutes and 18 minutes. If they ring together at 9:00 AM, when will they ring together again?

Answer key

Show suggested answers
  1. 252 = 105 x 2 + 42, 105 = 42 x 2 + 21, 42 = 21 x 2 + 0; HCF = 21.
  2. 96 = 2^5 x 3 and 404 = 2^2 x 101; HCF = 4, LCM = 9696.
  3. HCF of 45 and 75 = 15, LCM = 225, and 15 x 225 = 3375 = 45 x 75.
  4. Find HCF of differences 9377 - 1251, 15628 - 9377, and 15628 - 1251; the required number is their HCF.
  5. 3 x 5 x 7 + 7 = 7(15 + 1) = 112, so it is composite.
  6. 7 x 11 x 13 + 13 = 13(77 + 1), so it has factors other than 1 and itself.
  7. Assume sqrt(3) = p/q in lowest terms, then p^2 = 3q^2, so p and q are both divisible by 3, a contradiction.
  8. If 2 + sqrt(5) is rational, then sqrt(5) is rational after subtracting 2, which is false.
  9. LCM of 12, 15, 20, and 27 = 540.
  10. LCM of 12 and 18 = 36 minutes, so they ring together again at 9:36 AM.

How to use this worksheet

Students can solve the questions independently first, then review errors with a parent or teacher. Teachers can use it as a short class activity, homework sheet, or revision check.

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