Question

the digit in the ones place of 2^88 is 6. what is the digit in the ones place of 2^90?
a. 2
b. 4
c. 6
d. 8

Explanation:

Step1: Find the pattern of the ones digit of powers of 2

We know that:

• \(2^1 = 2\), ones digit is 2
• \(2^2 = 4\), ones digit is 4
• \(2^3 = 8\), ones digit is 8
• \(2^4 = 16\), ones digit is 6
• \(2^5 = 32\), ones digit is 2
• \(2^6 = 64\), ones digit is 4
• \(2^7 = 128\), ones digit is 8
• \(2^8 = 256\), ones digit is 6

So the pattern of the ones digit of powers of 2 repeats every 4: 2, 4, 8, 6, 2, 4, 8, 6,...

Step2: Determine the position of \(2^{90}\) in the pattern

First, we know that \(2^{88}\) has a ones digit of 6 (given). Then \(2^{89}=2^{88}\times2\), so the ones digit of \(2^{89}\) is \(6\times2 = 12\), so ones digit is 2. Then \(2^{90}=2^{89}\times2\), so the ones digit of \(2^{90}\) is \(2\times2 = 4\).

Alternatively, we can use the cycle. The cycle length is 4. We can find the remainder when 90 is divided by 4. \(90\div4 = 22\) with a remainder of 2. So the ones digit of \(2^{90}\) is the same as the ones digit of \(2^2\), which is 4.

Answer:

B. 4