The ratio of two numbers is 3:4 and their HCF is 4.Their LCM is:

Option 1 : 12 Option 2 : 16 Option 3 : 24

**Option 4 : 48**

**Let the numbers be 3x and 4x. Then, their H.C.F. = x. So, x = 4.**

**So, the numbers 12 and 16.**

**L.C.M. of 12 and 16 = 48.**

Ques 32 : Choose the correct answer.

A rectangular courtyard 3.78 meters long and 5.25 meters wide is to be paved exactly with square tiles ,all of the same size. What is the largest size of the tile which could be used for the purpose?

Option 1 : 14 cm **Option 2 : 21 cm** Option 3 : 42 cm Option 4 : None of these

**3.78 meters =378 cm = 2 × 3 × 3 × 3 × 7**

**5.25 meters=525 cm = 5 × 5 × 3 × 7**

**Hence common factors are 3 and 7**

**Hence LCM = 3 × 7 = 21**

**Hence largest size of square tiles that can be paved exactly with square tiles is 21 cm.**

Ques 33 : Choose the correct answer.

The least perfect square which is divisible by 3, 4, 5, 6, 8 is:

Option 1 : 900 Option 2 : 1200 Option 3 : 2500 **Option 4 : 3600**

**To find least perfect square number which is divisible by 3, 4, 5, 6 and 8.**

**LCM (3, 4, 5, 6 ,8) = 120**

**120 = 2 x 2 x 2 x 3 x 5.**

**As 2,3 and 5 are not in pair in LCM’s factor so we need to multiply 120 by 5 and 3,2 to make it a perfect square.**

**120 x 2 x 5 x 3 = 3,600.**

**∴ 3600 is the least perfect square divisible by 3, 4, 5, 6 and 8.**

Ques 34 : Choose the correct answer.

What will be obtained if 8 is subtracted from the HCF of 168, 189, and 231?

Option 1 : 15 Option 2 : 10 Option 3 : 21 **Option 4 : None of these**

**168 = 3 x 7 x 2 x 2 x 2 **

**189 = 3 x 7 x 3 x 3 **

**231 = 3 x 7 x 11 **

**As you can see, there is a column of 3 and 7. Consider these as one number, then multiply. 3 x 7 = 21 **

Ques 35 : Choose the correct answer.

The largest four digit number which is a multiple of 8, 10,12 and 15 is:

Option 1 : 120 Option 2 : 9600 Option 3 : 9840 **Option 4 : 9960**

**lcm=120**

**on dividing 9999 by 120 we get 39 as reminder **

**require no is =9999-39=9960**

Ques 36 : Choose the correct answer.

If logx (0.1) = -1/3, then the value of x is:

Option 1 : 10 Option 2 : 100 **Option 3 : 1000** Option 4 : 1/1000

Ques 37 : Choose the correct answer.

If ax = by, then:

Option 1 : log(a/b) = x/y Option 2 : log(a) / log(b) = x/y **Option 3 : log(a) / log(b) = y/x **Option 4 : None of these

Ques 38 : Choose the correct answer.

If log8 x + log8 (1/6) = 1/3 then the value of x is:

**Option 1 : 12 **Option 2 : 16 Option 3 : 18 Option 4 : 24

**log8 x + log8 (1/6) = 1/3**

**=> (log x/ log 8) + (log 1/6 / log 8) = log (81/3) = log 2**

**=> log x = log 2 – log 1/6 = log (2*6/1)= log 12**

Ques 39 : Choose the correct answer.

If log x + log y = log (x + y), then:

Option 1 : x = y Option 2 : xy=1 Option 3 : y = (x-1)/x **Option 4 : y = x/(x-1)**

**log (x) + log(y) = log(x + y) **

**=> log(xy) = log(x+y) **

**=> xy = x + y **

**=> xy – y = x **

**y(x – 1) = x **

**y = x /(x – 1)**

Ques 40 : Choose the correct answer.

If log10 7 = a, then log10(1/70) is equal to:

**Option 1 : -(1 + a)** Option 2 : (1 + a)-1 Option 3 : a/10 Option 4 : 1/10a

log10(1/70)= log10 1 – log10 70

= – log10 (7 x 10)=

– (log10 7 + log10 10)

= – (*a* + 1).