If x and y are positive integers such that 5^x \cdot 2^y = 10^{2x-1} \cdot 4^{x+1}, what i… — GRE General Test

If $x$ and $y$ are positive integers such that $5^x \cdot 2^y = 10^{2x-1} \cdot 4^{x+1}$ , what is the value of $y - x$ ?

The value of

y - x

is 4.

By prime-factorizing the bases on the right-hand side,

10^{2x-1} \cdot 4^{x+1}

becomes

(2 \cdot 5)^{2x-1} \cdot (2^2)^{x+1} = 5^{2x-1} \cdot 2^{4x+1}

. Matching the powers of 5 gives

x = 2x - 1

, which yields

x = 1

. Matching the powers of 2 gives

y = 4x + 1

, which yields

y = 5

. Subtracting

x

from

y

gives

5 - 1 = 4

Rewrite composite bases into prime factor bases on the right side of the equation.

10^{2x-1} = (2 \cdot 5)^{2x-1} = 2^{2x-1} \cdot 5^{2x-1}

and

4^{x+1} = (2^2)^{x+1} = 2^{2(x+1)} = 2^{2x+2}

Converting all terms to prime bases (2 and 5) allows equating corresponding exponents.

Combine terms with identical bases on the right side.

10^{2x-1} \cdot 4^{x+1} = 5^{2x-1} \cdot 2^{(2x-1) + (2x+2)} = 5^{2x-1} \cdot 2^{4x+1}

Applying the product rule of exponents

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

simplifies the right side.

Equate exponents of corresponding prime bases from both sides of

5^x \cdot 2^y = 5^{2x-1} \cdot 2^{4x+1}

Equating powers of 5 yields

x = 2x - 1 \implies x = 1

. Equating powers of 2 yields

y = 4x + 1

Since 2 and 5 are distinct prime numbers, their corresponding exponents must be equal.

Calculate

y

and evaluate

y - x

y = 4(1) + 1 = 5

, so

y - x = 5 - 1 = 4

Substituting

x = 1

gives

y = 5

, satisfying the final question requirement.

Decomposing exponential bases into prime factors and applying exponent rules (

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

and

(a^m)^n = a^{mn}

) to solve system equations of powers.

Question