Question

the problem (1,234)² means 1,234 1,234; (1,234)³ means 1,234 1,234 * 1,234; and so forth. when (1,234)²³ is completely multiplied out, what will the number be in the ones place?

Explanation:

Step1: Analyze the pattern of the units digit

When we look at the powers of \(4\) (since the units digit of \(1234\) is \(4\)):

• \(4^1 = 4\), units digit is \(4\).
• \(4^2=4\times4 = 16\), units digit is \(6\).
• \(4^3 = 4^2\times4=16\times4 = 64\), units digit is \(4\).
• \(4^4=4^3\times4 = 64\times4=256\), units digit is \(6\).

We can observe that for \(n = 1,2,3,4,\cdots\), if \(n\) is odd, the units digit of \(4^n\) is \(4\), and if \(n\) is even, the units digit of \(4^n\) is \(6\).

Step2: Determine the parity of the exponent

The exponent \(n = 23\). Since \(23\) is an odd number.

Answer:

The number in the ones place is \(4\).