start with 1, multiply by 2. then multiply the result by 3.then continue to alternate multiplying by 2 and by 3.what is the ones digit after the twelfth multiplication
2
6
12
36
72
216
see the pattern? it is either 2 or 6. The even number of multiplications are 6
multiply 2 by 8, then multiply 3 by the result
To find the ones digit after the twelfth multiplication, let's follow the sequence step-by-step:
Step 1: Start with 1.
Step 2: Multiply 1 by 2, resulting in 2.
Step 3: Multiply 2 by 3, resulting in 6.
Step 4: Multiply 6 by 2, resulting in 12.
Step 5: Multiply 12 by 3, resulting in 36.
Step 6: Multiply 36 by 2, resulting in 72.
Step 7: Multiply 72 by 3, resulting in 216.
Step 8: Multiply 216 by 2, resulting in 432.
Step 9: Multiply 432 by 3, resulting in 1296.
Step 10: Multiply 1296 by 2, resulting in 2592.
Step 11: Multiply 2592 by 3, resulting in 7776.
Step 12: Multiply 7776 by 2, resulting in 15552.
Now, let's focus on the ones digit of each step:
Step 1: 1
Step 2: 2
Step 3: 6
Step 4: 2
Step 5: 6
Step 6: 2
Step 7: 6
Step 8: 2
Step 9: 6
Step 10: 2
Step 11: 6
Step 12: 2
The ones digit after the twelfth multiplication is 2.
To solve this problem, we need to simulate the process of alternating multiplication by 2 and 3 and track the ones digit after the twelfth multiplication. Let's break it down step by step:
1. Start with the number 1.
2. Multiply it by 2: 1 * 2 = 2.
3. Multiply the result (2) by 3: 2 * 3 = 6.
4. Multiply the result (6) by 2: 6 * 2 = 12.
5. Multiply the result (12) by 3: 12 * 3 = 36.
6. Continue this alternating pattern of multiplying by 2 and by 3 for a total of twelve multiplications.
Now, instead of performing all twelve multiplications, we will focus on tracking the ones digit after each multiplication. The ones digit is the rightmost digit in a number.
Starting with 1, let's keep track of the ones digit after each multiplication:
1 * 2 = 2 (ones digit: 2)
2 * 3 = 6 (ones digit: 6)
6 * 2 = 12 (ones digit: 2)
12 * 3 = 36 (ones digit: 6)
...
We can see that the ones digit repeats itself in a pattern: 2, 6, 2, 6, and so on.
Since we need to find the ones digit after the twelfth multiplication, we can extend this pattern by calculating the ones digit accordingly:
Pattern: 2, 6, 2, 6, ...
12th multiplication: the twelfth number in the pattern is 2.
Therefore, the ones digit after the twelfth multiplication is 2.