Given a number, a strange calculator can only do the following: multiply it by 2 or by 3, or to
raise it to the power 2 or 3. Starting with the number 15, what can be obtained by applying
this calculator 5 times consecutively?
A) 2^8 . 3^5 . 5^6
B) 2^8 . 3^4 . 5^2
C) 2^3 . 3^3 . 5^3
D) 2^6 . 3^6 . 5^4
E) 2 . 3^2 . 5^6
To solve this problem, let's break it down step by step.
Starting with the number 15, we need to apply the calculator 5 times consecutively. We have four operations we can choose from at each step: multiply by 2, multiply by 3, raise to the power 2, or raise to the power 3.
Let's list out all the possible combinations of operations we can choose from at each step. We will use M2 for multiply by 2, M3 for multiply by 3, P2 for raise to the power 2, and P3 for raise to the power 3.
Step 1: 15 = M3 (multiply by 3)
Step 2: 45 = M3 (multiply by 3)
Step 3: 135 = M3 (multiply by 3)
Step 4: 405 = M3 (multiply by 3)
Step 5: 1215 = M3 (multiply by 3)
Now we have the final number after applying the calculator 5 times consecutively: 1215.
To determine which answer choice is correct, let's simplify 1215 using prime factorization.
Prime factorization of 1215:
1215 = 3^5 * 5^1 * 1^1
Comparing this with the answer choices, we can see that the correct answer is:
A) 2^8 * 3^5 * 5^6
So, the answer is A) 2^8 * 3^5 * 5^6.