The last digit of the number (32)³² is

Solution:

1. Focus on the units digit of the base:
   The number 32 ends in 2, so the calculation depends on the units digit of 2³².
   Essentially, we only need to determine the units digit of 2³².

2. Identify the pattern in powers of 2:
   Observe the units digits for the first few powers of 2:

   • 2¹ = 2 → Units digit is 2
   • 2² = 4 → Units digit is 4
   • 2³ = 8 → Units digit is 8
   • 2⁴ = 16 → Units digit is 6
   • 2⁵ = 32 → Units digit is 2 (pattern repeats)

   The units digit repeats every 4 powers: 2, 4, 8, 6.

3. Find the position of 2³² in the pattern:
   Divide the exponent 32 by 4:
   32 ÷ 4 = 8 (remainder 0).
   A remainder of 0 means 2³² corresponds to the last digit in the pattern, which is 6.
4. Conclude for (32)³²:
   Since the units digit of 2³² is 6, the units digit of (32)³² is also 6.

Final Answer:

The last digit of (32)³² is 6.