**1.**

**Express 140 as a product of its prime factors**

**2.**

**Find the LCM and HCF of 12, 15 and 21 by the prime factorization method.**

**3.**

**Find the LCM and HCF of 6 and 20 by the prime factorization method.**

**4.**

**State whether13/3125 will have a terminating decimal expansion or a non-terminating repeating decimal.**

**5.**

**State whether 17/8 will have a terminating decimal expansion or a non-terminating repeating decimal.**

**6.**

**Find the LCM and HCF of 26 and 91 and verify that LCM × HCF = product of the two numbers.**

**7.**

**Use Euclid’s division algorithm to find the HCF of 135 and 225**

**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.**

**Prove that √3 is irrational.**

**10.**

**Show that 5 – √3 is irrational**

**11.**

**Show that any positive odd integer is of the form 6**

*q*+ 1, or 6*q*+ 3, or 6*q*+ 5, where*q*is some integer.**12.**

**An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?**

**13.**

**Express 156 as a product of its prime factors.**

**14.**

**Find the LCM and HCF of 17, 23 and 29 by the prime factorization method.**

**15.**

**Find the HCF and LCM of 12, 36 and 160, using the prime factorization method.**

**16.**

**State whether 6/15 will have a terminating decimal expansion or a non-terminating repeating decimal.**

**17.**

**State whether35/50 will have a terminating decimal expansion or a non-terminating repeating**

**18.**

**decimal.**

**19.**

**Find the LCM and HCF of 192 and 8 and verify that LCM × HCF = product of the two numbers.**

**20.**

**Use Euclid’s algorithm to find the HCF of 4052 and 12576.**

**21.**

**Show that any positive odd integer is of the form of 4q + 1 or 4q + 3, where q is some integer.**

**22.**

**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.**

**23.**

**Prove that 3√2 5 is irrational.**

**24.**

**11. Prove that 1/√2 is irrational. (3 marks)**

**25.**

**In a school there are tow sections- section A and Section B of class X. There are 32 students in section A and 36 students in section B. Determine the minimum number of books required for their class library so that they can be distributed equally among students of section A or section B.**

**26.**

**Express 3825 as a product of its prime factors.**

**27.**

**Find the LCM and HCF of 8, 9 and 25 by the prime factorization method.**

**28.**

**Find the HCF and LCM of 6, 72 and 120, using the prime factorization method.**

**29.**

**State whether 29/343 will have a terminating decimal expansion or a non-terminating repeating decimal.**

**30.**

**State whether 23/ 23 52will have a terminating decimal expansion or a non-terminating repeating decimal**

**31.**

**Find the LCM and HCF of 336 and 54 and verify that LCM × HCF = product of the two numbers**

**32.**

**Use Euclid’s division algorithm to find the HCF of 867 and 255**

**33.**

**Show that every positive even integer is of the form 2**

*q*, and that every positive odd integer is of the form 2*q*+ 1, where*q*is some integer.**34.**

**Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m,9lm + 1 or 9m + 8.**

**35.**

**Prove that 7 √5 is irrational.**

**36.**

**Prove that √5 is irrational.**

**37.**

**There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, 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?**

**38.**

**Express 5005 as a product of its prime factors.**

**39.**

**Find the LCM and HCF of 24, 36 and 72 by the prime factorization method.**

**40.**

**Find the LCM and HCF of 96 and 404 by the prime factorization method**

**41.**

**State whether 64/455 will have a terminating decimal expansion or a non-terminating repeating decimal**

**42.**

**State whether15/ 1600 will have a terminating decimal expansion or a non-terminating repeating decimal.**

**43.**

**Find the LCM and HCF of 510 and 92 and verify that LCM × HCF = product of the two numbers.**

**44.**

**Use Euclid’s division algorithm to find the HCF of 196 and 38220 (3 marks)**

**45.**

**Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m,9m + 1 or 9m + 8**

**46.**

**Show that every positive odd integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer**

**47.**

**10. Show that 3√ 2 is irrational.**

**48.**

**Prove that 3 + 2 √5 is irrational.**

**49.**

**A sweet seller has 420 kaju barfis and 130 badam barfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the maximum number of barfis that can be placed in each stack for this purpose?**

**50.**

**Use Euclid’s division algorithm to find the HCF of : (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255**

**51.**

**Show that any positive odd integer is of the form 6**

*q*+ 1, or 6*q*+ 3, or 6*q*+ 5, where*q*is some integer.**52.**

**An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march? Sol. Hints: Find the HCF of 616 and 32**

**53.**

**Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3**

*m*or 3*m*+ 1 for some integer*m*. [Hint : Let*x*be any positive integer then it is of the form 3*q*, 3*q*+ 1 or 3*q*+ 2. Now square each of these and show that they can be rewritten in the form 3*m*or 3*m*+ 1.]**54.**

**Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9**

*m*, 9*m*+ 1 or 9*m*+ 8.**55.**

**Consider the numbers 4**

*n***, where**

*n*is a natural number. Check whether there is any value of*n*for which 4

*n***ends with the digit zero.**

**56.**

**Find the LCM and HCF of 6 and 20 by the prime factorization method.**

**57.**

**Find the HCF of 96 and 404 by the prime factorization method. Hence, find their LCM.**

**58.**

**Find the HCF and LCM of 6, 72 and 120, using the prime factorization method.**

**59.**

**Find the value of y if the HCF of 210 and 55 is expressible in the form 210 x 5 + 55y**

**60.**

**Prove that no number of the type 4K + 2 can be a perfect square.**

**61.**

**Express each number as a product of its prime factors:**

**62.**

**140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429**

**63.**

**Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.**

**64.**

**26 and 91 (ii) 510 and 92 (iii) 336 and 54**

**65.**

**Find the LCM and HCF of the following integers by applying the prime factorization method.**

**66.**

**12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25**

**67.**

**Given that HCF (306, 657) = 9, find LCM (306, 657).**

**68.**

**Check whether 6n can end with the digit 0 for any natural number n.**

**69.**

**Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.**

**70.**

**There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, 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?**

**71.**

**Use Euclid’s division lemma to show that the square of any positive integer is of the form 5q, 5q+1,5q+4 for some integer q.**

**72.**

**Show that any one of the numbers (n + 2), n and (n + 4) is divisible by 3.**

**73.**

**If 793800 = 2 3 x 3 m x 5 n x 7 2, find the value of m and n.**

**74.**

**If the HCF of 210 and 55 is expressible in the form 210 × 5 + 55y then find y.**

**75.**

**What will be the value of 0.**

**+ 0.**

**?**

**76.**

**Given that HCF (135, 225) = 45. Find LCM (135, 225).**

**77.**

**Solve √18x√50. What type of number is it, rational or irrational?**

**78.**

**What type of decimal expansion will69/60 represent? After how many places will the decimal expansion terminate?**

**79.**

**Find the H.C.F. of the smallest composite number and the smallest prime number.**

**80.**

**If a = 4q + r then what are the conditions for a and q. What are the values that r can take?**

**81.**

**What is the smallest number by which √5 - √ 3 be multiplied to make it a rational no? Also find the no. so obtained.**

**82.**

**What is the digit at unit’s place of 96?**

**83.**

**Find one rational and one irrational no. between √3 and √5.**

**84.**

**If the no. pn never to end with the digit 0 then what are the possible value (s) of p?**

**85.**

**State Euclid’s Division Lemma and hence find HCF of 16 and 28.**

**86.**

**State fundamental theorem of Arithmetic and hence find the unique fraternization of 120.**

**87.**

**Prove that 1/(2 - √5) is irrational number.**

**88.**

**Check whether 5 × 7 × 11 + 6 is a composite number.**

**89.**

**Check whether 7 × 6 × 3 × 5 + 5 is a composite number.**

**90.**

**If the H C F of 657 and 963 is expressible in the form of 657x + 963**

**x**

**- 15 find x. (Ans:x=22)**

**91.**

**Express the GCD of 48 and 18 as a linear combination. (Ans: Not unique)**

**92.**

**Prove that one of every three consecutive integers is divisible by 3.**

**93.**

**Find the largest possible positive integer that will divide 398, 436, and 542 leaving remainder 7, 11, 15 respectively. (Ans: 17)**

**94.**

**Find the least number that is divisible by all numbers between 1 and 10 (both inclusive).**

**95.**

**Show that 571 is a prime number.**

**96.**

**If d is the HCF of 30, 72, find the value of x & y satisfying d = 30x + 72y. (Ans:5, -2 (Not unique)**

**97.**

**Show that the product of 3 consecutive positive integers is divisible by 6.**

**98.**

**Find the greatest number of 6 digits exactly divisible by 24, 15 and 36.(Ans:999720)**

**99.**

**If a and b are positive integers. Show that**

**Ö**

**2 always lies between a/b and (a-2b)/(a+b)**

**100.**

**Prove that (**

**√**

*n***-**

**1**

**+**

**√**

*n***+**

**1 ) is irrational, for every n**

**Î**

**N**