# Class 10 – Real Numbers – Important Questions

Real Numbers – Important Questions

1. HCF of 65 and 117 is expressible in the form 65n – 117, then find the value of n.
2. On a morning walk, three persons step out together and their steps measure 30 cm, 36 cm and 40 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?
3. Express 429 as a product of its prime factors.
4. Find after how many places of decimal the decimal form of the number $\frac{27}{{2^{3}}.5^{4}.3^{2}}$ will terminate.
5. Prove that $\sqrt{2}$ is an irrational number.
6. Find the largest number which on dividing 1251, 9377 and 15628 leaves remainder 1, 2 and 3 respectively.
7. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
8. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.
9. Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer.
10. Consider the numbers 4n, where n is a natural number. Check whether there is any value of n for which 4n ends with the digit zero.
11. Given that HCF (306, 657) = 9, find LCM (306, 657).