Question

125 is multiplied by either 10 or 15. The resultant number is again multiplied by either 10 or 15. This process continues.
Which of the following CANNOT be a resultant number at any point in time?

• A. \( 2^{235} \times 3^{453} \times 5^{691} \)
• B. \( 2^{235} \times 3^{8245} \times 5^{1080} \)
• C. \( 2^{476} \times 3^{455} \times 5^{1034} \)
• D. \( 2^{689} \times 3^{512} \times 5^{1145} \)
• E. \( 2^{689} \times 3^{912} \times 5^{1604} \)

Hint:

In problems involving prime factorizations, always track how the factors change with each operation. Ensure that the powers of the factors align with the rules of multiplication.

Answer 1

The Correct Option is D

Step 1: Understand the process.
In each step, the number is multiplied by either 10 or 15. The prime factorization of 10 is \( 2 \times 5 \), and the prime factorization of 15 is \( 3 \times 5 \). Thus, with each multiplication, the powers of 2, 3, and 5 in the prime factorization of the resultant number will change.
Step 2: Examine the changes in the prime factorization.
Each time we multiply by 10, we increase the power of 2 and the power of 5 by 1. Each time we multiply by 15, we increase the power of 3 and the power of 5 by 1. Therefore, the resultant number will always have the form \( 2^a \times 3^b \times 5^c \), where \( a \), \( b \), and \( c \) are non-negative integers, and \( c \) will always be greater than or equal to \( a + b \) because of the repeated multiplication by 5.
Step 3: Check the given options.
For option (D), the power of 5 is smaller than the power of 2 and 3, which is not possible because the power of 5 should be greater than or equal to the sum of the powers of 2 and 3. Hence, this option is not possible.
Step 4: Conclusion.
The correct answer is (D).