**Q 1 – The HCF of two numbers is 27 and their LCM is 162. If one of the numbers is 54, then the other number is(a) 36**

**(b) 35**

**(c) 9**

**(d) 81**

**Q 2 – 225 can be expressed as**

**(a) an even number**

**(b) an odd number**

**(c) an odd prime number**

**(d) a prime number**

**Q 4. ( n^{2 }–1) is divisible by 8 if n is**

**(a) an integer**

**(b) a natural number**

**(c) an odd integer**

**(d) an even integer**

**Q 5 – Assertion: The HCF of two numbers is 5 and their product is 150, then their LCM is 30.**

**Reason : For any two positive integers a and b, HCF (a, b) + LCM (a, b) = a** x

*b*.**(a) Both assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of assertion (A).**

**(b) Both assertion (A) and Reason (R) are true but reason (R) is not the correct explanation of assertion (A).**

**(c) Assertion (A) is true but reason (R) is false.**

**(d) Assertion (A) is false but reason (R) is true.**

**Q 6 – Find the HCF and LCM of 90 and 144 by the method of prime factorization.**

**Q 7 – Find the missing numbers a, b, c and d in the given factor tree:**

**Q 8 – The length, breadth and height of a room are 8 m 50 cm, 6 m 25 cm and 4 m 75 cm respectively. Find the length of the longest rod that can measure the dimensions of the room exactly.**

**Q 9 – Show that 5** √**6 is an irrational number.**

**Q 10 – An army contingent of 612 members is to march behind an army band of 48 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?**

**Q 11 – Prove that 2 + 5** √**3 is an irrational number, given that 3 is an irrational number.**

**Q 12 – Show that numbers 8 ^{n} can never end with digit 0 of any natural number n.**

**Q 14 – If p is prime number, then prove that**

**√p is an irrational.**

**Q 15 – Prove that n ^{2}– n is divisible by 2 for every positive integer n.**