power of 2 ends on:
2 - 4 -8 - 6 and repeat

so 2^24 will repeat this endings 6 times and end on 6
and divided by 5 will result in reminder 1

answer A

Option A

Remainder ( \(\frac{12^{24}}{5}\)) = ?

Remainder ( \(\frac{12^{24}}{5}\)) = Remainder (unit's digit of (12^{24})/5)

: (\(12^{24}\)) = (\(2^{48} * 3^{24}\))
: Unit's digit (\(2^{48} * 3^{24}\)) = ( 6 * 1) [Using power cyclicity of 2 & 3] = 6

:Remainder (\(\frac{6}{5}\)) = 1

12^24 can be written as (10+2)^24.

By Binomial expansion, we know that all the terms will have multiples of 10 except the last term which is 2^24. So, the remainder will depend upon 2^24.

Now, to find out the last digit of 2^24 :
2^1=2
2^2=4
2^3=8
2^4=16
2^5=32

As we can see the units digit start to repeat after 4, that means cyclicity is 4.
Dividing the power of 2, i.e, 24 by 4 we get 0.
That means the last digit of 2^24 will be same as last digit of 2^4 which is 6.
Hence, when divided by 5 the remainder will be 1.

Ans: A

\(12 = \frac{(10 + 2)}{5}\) ( Remainder 2 )
\(12^2 = \frac{(10 + 2)^2}{5}\) ( Remainder 4 )
\(12^3 = \frac{(10 + 2)^3}{5}\) ( Remainder 3 )
\(12^4 = \frac{(10 + 2)^4}{5}\) ( Remainder 1 )

Now, \(12^{24} = 12^4*6\)

Since, \(12^4\) will have a remainder of 1 ; \(12^{24}\) will also have a remainder of 1 when divided by 5

Thus, answer must be (A) 1
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When solving this problem, we should recall the rule that we can determine the remainder when a number is divided by 5 by simply dividing the units digit of that number by 5. Thus, to determine the remainder when 12^24 is divided by 5, we need to first calculate the units digit of 12^24.

Since we only care about the units digit, we can evaluate the pattern of units digits for 2^n for positive integer values of n. That is, let’s look at the pattern of the units digits of powers of 2, which will result in the same pattern as 12^n. When writing out the pattern, notice that we are ONLY concerned with the units digit of 2 raised to each power.

2^1 = 2

2^2 =4

2^3 = 8

2^4 = 6

2^5 = 2

The pattern of the units digit of powers of 2 repeats every 4 exponents. The pattern is 2–4–8–6. In this pattern, all positive exponents that are multiples of 4 will produce a 6 as their units digit. Thus:

2^24 has a units digit of 6.

Finally, since the remainder is 1 when 6 is divided by 5, the remainder is also 1 when 12^24 is divided by 5.

Answer: A
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\(\frac{12^{24}}{5}\)

\(=\frac{(10 + 2) ^{24}}{5}\)

\(\frac{10}{5}\) will be completely divisible and leave no remainder.....

\(\frac{2^4}{5}\) will leave remainder 1

Hence, \(\frac{2^{24}}{5} = \frac{2^{4*6}}{5}\) will leave remainder 1....

Hence, Answer must be (A) 1...
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