Download 2√45 +3√20 2√5

Summer Vacation Assignment
1. If
𝑝
𝑞
𝑝
𝑞
is a rational number (q ≠ 0), what is the condition of q so that the decimal representation of
is terminating?
2. Write a rational number between √2 and √3
3. The HCF of 45 and 105 is 15. Write their LCM.
4. Write whether
2√45 +3√20
2√5
on simplification gives a rational number or an irrational number.
5. If HCF(6, a) = 2 and LCM(6, a) = 60, then find the value of a.
6. If HCF(72, 120) = 24, then find LCM(72, 120).
7. If HCF and LCM of two natural numbers are 12 and 180 and one of the numbers is 36 then find
the other number.
8.
If HCF(2520, 6600) = 120 and LCM(2520, 6600) = 252k, then find the value of k.
9. Find the product of HCF and LCM of the smallest prime number and the smallest composite
number.
10. If two positive integers a and b are written as a = x3y2 and b = xy3, where x and y are prime
numbers, find HCF(a, b) and LCM(a,b).
11. If the HCF of 65 and 117 is expressible in the form 65 m – 117 then find the value of m.
12. Prove that if x and y are odd positive integers, x2 + y2 is even but not divisible by 4.
13. Find the largest number that divides 2053 and 967 and leaves a remainder of 5 and 7 respectively.
14. Find the largest number that will divide 398, 436 and 542 leaving remainders 7, 11 and 15
respectively.
15. Find the HCF of 81 and 237 and express it as 81x + 237y.
16. Prove that √2 is an irrational number.
17. Prove that √5 is an irrational number.
18. Prove that 5 − √3 is an irrational number.
19. Prove that √2 + √5 is an irrational number.
20. 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.
21. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m,
9m + 1 or 9m + 8.
22. Show that 4n cannot end with the digit 0 for any natural number n.