If a and b are positive integers such that 2^{a+3} \cdot 5^b - 2^a \cdot 5^{b+1} = 6,000, … — GMAT

If $a$ and $b$ are positive integers such that $2^{a+3} \cdot 5^b - 2^a \cdot 5^{b+1} = 6,000$ , what is the value of $a \cdot b$ ?

Factoring out

2^a \cdot 5^b

yields

2^a \cdot 5^b (2^3 - 5^1) = 3 \cdot 2^a \cdot 5^b = 6,000

. Dividing by 3 yields

2^a \cdot 5^b = 2,000

. Expressing

2,000

in prime factored form gives

2^4 \cdot 5^3

. Matching exponents for the prime bases gives

a = 4

and

b = 3

. The product

a \cdot b

4 \times 3 = 12

Rewrite the terms in the expression using exponent rules to isolate common bases.

2^{a+3} \cdot 5^b - 2^a \cdot 5^{b+1} = (2^a \cdot 2^3) \cdot 5^b - 2^a \cdot (5^b \cdot 5^1)

Applying

x^{m+n} = x^m \cdot x^n

allows us to extract common powers of

2^a

and

5^b

Factor out the common term

2^a \cdot 5^b

from the left side of the equation.

2^a \cdot 5^b (2^3 - 5^1) = 2^a \cdot 5^b (8 - 5) = 3 \cdot 2^a \cdot 5^b

Simplifying the constant factor in parentheses simplifies the equation.

Divide both sides of the equation by 3 and perform prime factorization on the resulting integer.

3 \cdot 2^a \cdot 5^b = 6,000 \implies 2^a \cdot 5^b = 2,000 = 2^4 \cdot 5^3

Prime factorization of

2,000

determines the unique integer exponents for bases 2 and 5.

Equate the corresponding exponents and calculate the requested product

a \cdot b

a = 4

and

b = 3 \implies a \cdot b = 4 \times 3 = 12

Since 2 and 5 are prime numbers, the prime factorization representation is unique.

Factoring Exponents and Prime Factorization

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