Question

L.C.M. of the given two numbers is double the greater number. And the difference

between smaller number and G.C.F. is 4. Therefore the smaller number is …………

(A) 8 (B) 6 (C) 12 (D) 16​

Answer 1

(a) 8 is the right answer

Let (x,y)∈N2,x<y(x,y)∈N2,x<y be our ordered pair of two integers.

As x−4x−4 is their greater common divisor,

∃k∈N∗,k(x−4)=y⇒x−4>0⇒x>4(1)(1)∃k∈N∗,k(x−4)=y⇒x−4>0⇒x>4

As their lower common multiple is 2y2y

∃m∈N∗,mx=2y(2)(2)∃m∈N∗,mx=2y

Thus

2k(x−4)=mx⇔(2k−m)x=82k(x−4)=mx⇔(2k−m)x=8

xx has to be a divisor of 88 , so we know that x∈{1,2,4,8}x∈{1,2,4,8}

But, from (1)(1) we know that x∈[5,∞)x∈[5,∞) , so

x∈{1,2,4,8}∩[5,∞)⇔x=8x∈{1,2,4,8}∩[5,∞)⇔x=8

Answer 2

Given:

Two numbers = x and y and x > y

LCM = 2x

GCF = y - 4

To Find:

The smallest number

Solution:

LCM × GCF = Product of the numbers

LCM × GCF = x × y

2x × ( y -4 ) = xy

2xy - 8x - xy = 0

xy - 8x = 0

x ( y - 8) = 0

x = 0 and y = 8

Now, the value of x can not be zero, thus -

HCF = y - 4

= 8 - 4 = 4

Answer: The smallest number is 8.