# Real Numbers

• Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b. This result is known as Euclid’s division lemma.
• An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.
• A lemma is a proven statement used for proving another statement.
• HCF of two positive integers a and b is the largest positive integer d that divides both a and b.
• Euclid’s Division Algorithm : To obtain the HCF of two positive integers, say c and d with c > d, we follow the steps below :
• Step 1. Apply Euclid’s division lemma to find q and r where c = dq + r, 0 ≤ r < d.
• Step 2. If r = 0, then, d is the HCF of c and d. If r ≠ 0, then apply Euclid’s division lemma to d and r.
• Step 3. Continue this process till the remainder is zero. The divisor at this stage will be the required HCF.
• The Fundamental Theorem of Arithmetic:Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur. Or the prime factorisation of a natural number is unique, except for the order of its factors.
• Any number which cannot be expressed in the form $\dpi{120} \frac{p}{q}$ where p, and q are integers and q ≠ 0 is called an irrational number.
• Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer.
• The sum or difference of a rational and an irrational number is irrational.
• The product and quotient of a non-zero rational number and an irrational number is irrational.
• Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form $\dpi{120} \frac{p}{q}$ ,where p and q are coprime and the prime factorisation of q is of the form 2n5m, where n and m are non negative integers.
• Let x = $\dpi{120} \frac{p}{q}$  be a rational number such that the prime factorisation of q is of the form 2n5m, where n and m are non negative integers. Then, x has a decimal expansion which terminates.
• If x = $\dpi{120} \frac{p}{q}$  is a rational number, such that the prime factorisation of q is of the form 2m5n, where m and n are whole numbers. If m = n, then the decimal expansion of x will terminate after m places of decimal. If m > n, then the decimal expansion of x will terminate after m places of decimal. If n > m, then the decimal expansion of x will terminate after n places of decimal.
• Let x = $\dpi{120} \frac{p}{q}$ be a rational number, such that the prime factorisation of q is not of the form 2n5m, where n and m are non negative integers. Then x has a decimal expansion which is non terminating repeating (recurring).
• The decimal expansion of every rational number is either terminating or non-terminating repeating.

# Important Questions

1. Use Euclid’s division algorithm to fnd the HCF of 10224 and 9648.
2. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
3. The values of the remainder r, when a positive integer a is divided by 3, are 0 and 1 only. Is it true? Justify your answer.
4. Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is a positive integer.
5. Show that any positive even integer is of the form 6m, 6m + 2 or 6m + 4. where m is some integer.
6. Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.
7. Use Euclid’s Lemma to show that square of any positive integer is of form 4m or 4m + 1 for some integer m.
8. Prove that n2 – n is divisible by 2 for every positive integer n.
9. Use Euclid division lemma to show that cube of any positive integer is either of the form 9m, 9m + 1, or 9m + 8.
10. Prove that √2 is an irrational number.
11. Prove that √5 is an irrational number.
12. Prove that 2 + 3√2 is irrational.
13. Prove that  $\dpi{120} \frac{5\sqrt{2}}{3}$  is an irrational number.
14. If a is a non-zero rational and √b is irrational, then show that a√b is an irrational.
15. Show that 9n can’t end with 2 for any integer n.
16. Check whether 6n can end with the digit 0, for any natural number n.
17. A merchant has 120 litres of oil of one kind, 180 litres of another kind and 240 litres of third kind. He wants to sells the oil by flling, the three kinds of oil in tins of equal capacity. What should be the greatest capacity of such a tin?
18. There is a circular path around a sports feld. Sonia takes 18 minutes to drive one round of the feld, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?
19. On a morning walk, three boys step off together and their steps measure 45 cm, 40 cm and 42cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?
20. Find the largest number which can divide 1001, 1287 and 1573.
21. Find the smallest number divisible by 115, 138 and 161.
22. Find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3 respectively.
23. Find the smallest number which when divided by 161, 207 and 184 leaves remainder 21 in each case.
24. Without actually performing the long division, state whether the following number has a terminating decimal expansion or non terminating recurring decimal expansion $\dpi{120} \small \frac{543}{225}$ .
25. Write the denominator of the rational number $\dpi{120} \small \frac{257}{5000}$ in the form 2m × 5n, where m, n are non-negative integers. Hence, write its decimal expansion, without actual division.