Question

The sum of LCM and HCF of two numbers is 854. If the LCM is 60 times the HCF and one of the numbers is 70, then the other number is:

• A. 160
• B. 164
• C. 168
• D. 172

Answer 1

The Correct Option is C

To solve this problem, let's define the variables and equations based on the given information: 

Let \( H \) be the HCF of the two numbers and \( L \) be their LCM. From the problem, we know:

• \( L = 60H \)
• \( L + H = 854 \)

Substitute \( L = 60H \) into the second equation:

\( 60H + H = 854 \)

\( 61H = 854 \)

Step 1: Calculate HCF (H)

Solve for \( H \):

\( H = \frac{854}{61} = 14 \)

Step 2: Calculate LCM (L)

Now find \( L \) using \( L = 60H \):

\( L = 60 \times 14 = 840 \)

Step 3: Determine the other number

Let the two numbers be \( a = 70 \) and \( b \). For two numbers, the relationship between their LCM and HCF is:

\( a \times b = L \times H \)

Thus, \( 70 \times b = 840 \times 14 \)

Solving for \( b \):

\( 70b = 11760 \)

\( b = \frac{11760}{70} \)

\( b = 168 \)

The other number is therefore 168.

Answer 2

Let the two numbers be \(a\) and \(b\). We are given that \(a=70\). 

We are also given that LCM + HCF = 854 and LCM = 60 \(\times\) HCF.

Substituting LCM = 60 \(\times\) HCF into the first equation, we get 60 * HCF + HCF = 854, which simplifies to 61 * HCF = 854.

Therefore, HCF = \( \frac{854}{61} = 14 \).

Now, LCM = 60 \(\times\) 14 = 840.

We know that the product of two numbers is equal to the product of their LCM and HCF. So, \(a \times b = LCM \times HCF\).

\(70 \times b = 840 \times 14\)

\(b = \frac{840 \times 14}{70} = \frac{840}{5} = 168\)