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Real Numbers SHORT ANSWER TYPE QUESTIONS

Real Numbers

SHORT ANSWER TYPE QUESTIONS                                         [2 Marks]


1. Show that every positive even integer is of the from 2m, and that every positive odd integer is of the form 2m + 1, where m is some integer.
2. Show that any positive odd integer is of form 4m + 1 or 4m + 3, where m is some integer.
3. Show that any positive odd integer is of the form 6m + 1, or 6m + 3, or 6m + 5, where m is some integer.
4. Find the HCF of 1656 and 4025 by Euclid’s method.
5. Factorise 34650 using factor tree.
6. Find the HCF of 255 and 867 by prime factorisation.
7. Find the largest number which can divide 3528 and 2835.
8. Find the LCM of 2520 and 2268 by prime factorisation.
9. Find the smallest number which is divisible by 85 and 119.
10. Show that 5 - Ö3  is irrational.
11. Show that 3Ö2 is irrational.
12. Show that 1 / Ö2 is irrational.
13. Write the denominator of the rational number 257 / 5000 in the form 2m × 5n, where m, n are non-negative integers. Hence, write its decimal expansion, without actual division.
14. 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.
15. Two tankers contain 850 litres and 680 litres of petrol respectively. Find the maximum capacity of a container which can measure the petrol of either tanker in exact number of times.
16. Show that the sum and product of two irrational numbers (5 + Ö2) and (5 – Ö2) are rational numbers.
17. Without actually performing the long division, find if 987 / 10500 will have terminating or non-terminating repeating decimal expansion. Give reason for your answer.
18. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
19. Show that any positive integer is of the form 3q or 3q + 1 or 3q + 2 for some integer q.
20 Can the number 6n, n being a natural number, end with the digit 5? Give reasons.
21. Use Euclid’s division lemma to show that square of any positive integer is either of form 3m or 31m+ for some integer m.
22. Find the L.C.M. of 120 and 70 by fundamental theorem of Arithmetic.
23. Write 60 in form of  factor tree.
24. Without actually performing the long division, state whether the following number has a terminating decimal expansion or non terminating recurring decimal expansion 543 /  225.
25. Use Euclid’s division algorithm to find HCF of 870 and 225.
26. Check whether 6n can end with the digit 0, for any natural number n.
27. Explain why 11 × 13 × 15 × 17 + 17 is a composite number.
28. 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.
29. Check whether 15n can end with digit zero for any natural number n.
30. Find the LCM of 336 and 54 by prime factorisation method.  
31. Find the LCM and HCF of 120 and 144 by fundamental theorem of arithmetic.
32. Use Euclid’s Lemma to show that square of any positive integer is of form 4m or 41m+ for some integer m.
33. Using fundamental theorem of arithmetic, find the HCF of 26, 51 and 91
34. Find the LCM and HCF of 15, 18, 45 by the prime factorisation method..
35. Find the HCF and LCM of 306 and 54. Verify that HCF × LCM = Product of the two numbers




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