If A = \({128^{{{128}^{128}}}}\) then which of the following statemen

1.	If A = \({128^{{{128}^{128}}}}\) then which of the following statement regarding A is incorrect?Statement 1: A is divisible by 2Statement 2: A is divisible by 3Statement 3: A is divisible by 61. Statement I2. Statement II3. Statement III4. None of these
Answer» Correct Answer - Option 4 : None of these Given: The given expression is A = \({128^{{{128}^{128}}}}\) Concept Used: Basic simplification and concept of unit digit Calculation: The given expression can be written as 128^{128 × 128 × 128………… × 128 times} Again the expression can be simplified as 128^{128 × (128 × 128 × 128………… × 127 times)} ∴ 128^{128 × (128 × 128 × 128………… × 127 times)} = 128^{4 × 32 × (128 × 128 × 128………… × 127 times)} Now, let 32 × (128 × 128 × 128………… × 127 times) = m ∴ 128^{4 × 32 × (128 × 128 × 128………… × 127 times)} = 128^{4 × m} Now, unit digit of 128^{4 × m} ∴ 128^{4 × m} = (8)^{4 × m} For any value of n 8^{4 × m} always gives the unit digit as 6 ∴ 6 is divisible by 2 and 3 both and we know that the number divisible by 2 and 3 both will also divisible by 6 Hence, option (4) is correct

If A = \({128^{{{128}^{128}}}}\) then which of the following statement regarding A is incorrect?Statement 1: A is divisible by 2Statement 2: A is divisible by 3Statement 3: A is divisible by 61. Statement I2. Statement II3. Statement III4. None of these

Answer» Correct Answer - Option 4 : None of these

Given:

The given expression is A = \({128^{{{128}^{128}}}}\)

Concept Used:

Basic simplification and concept of unit digit

Calculation:

The given expression can be written as 128^{128 × 128 × 128………… × 128 times}

Again the expression can be simplified as 128^{128 × (128 × 128 × 128………… × 127 times)}

∴ 128^{128 × (128 × 128 × 128………… × 127 times)} = 128^{4 × 32 × (128 × 128 × 128………… × 127 times)}

Now, let 32 × (128 × 128 × 128………… × 127 times) = m

∴ 128^{4 × 32 × (128 × 128 × 128………… × 127 times)} = 128^{4 × m}

Now, unit digit of 128^{4 × m}

∴ 128^{4 × m} = (8)^{4 × m}

For any value of n 8^{4 × m} always gives the unit digit as 6

∴ 6 is divisible by 2 and 3 both and we know that the number divisible by 2 and 3 both will also divisible by 6

Hence, option (4) is correct