Question

If the G.C.D. of 4850 and 8730 is \(2^a5^b7^c\), then \(3a+2b+5c=\)

• A. 8
• B. 11
• C. 10
• D. 15

Hint:

To find the G.C.D. using prime factorization:
• Factorize both numbers completely.
• Take only the common prime factors.
• Choose the smallest exponent of each common prime.

Answer 1

The Correct Option is C

Concept: The Greatest Common Divisor (G.C.D.) of two numbers is the product of the common prime factors having the smallest powers.

Step 1: Find the prime factorization of each number.
For \(4850\), \[ 4850=485\times10 =5\times97\times2\times5 =2\times5^2\times97 \] For \(8730\), \[ 8730=873\times10 =3\times291\times2\times5 =3\times3\times97\times2\times5 =2\times3^2\times5\times97 \]

Step 2: Find the G.C.D.
The common prime factors are: \[ 2,\;5,\;97 \] Hence, \[ \gcd(4850,8730)=2\times5\times97 =970 \] Given, \[ 970=2^a5^b97^c \] Comparing powers, \[ a=1,\qquad b=1,\qquad c=1 \]

Step 3: Calculate the required value.
\[ 3a+2b+5c =3(1)+2(1)+5(1) =3+2+5 =10 \]

Step 4: Final conclusion.
Therefore, \[ \boxed{3a+2b+5c=10} \]