Question

◆Check whether (26)^n can end with the digit 5 for any n € N

◆Check whether 2^n can end with the digit 6 for any n€N

◆Find the units place digit in the expansion of the following -
$({191})^{1009}$
Kindly give correct ans .

Answer 1

Hey mate !!

Here's the answer !!

First let me solve one by one.

So taking the first case of $26^{n}$ ending with 5

So we know that for a number to end with 5, the prime factorisation of a number must have 5 as it's prime factor.

So calculating prime factorisation of 26 we get,

26 = 13 × 2

Since 26 has no prime factor 5 in it's prime factorisation, there is no natural number ' n ' for which $26^{n}$ will end with zero. This is guaranteed by the uniqueness of Fundamental Theroem Of Arithmetic.

Second case : $2^{n}$ ending with 6

For a number to end with 6, the prime factorisation of that number must be having both 2. But the prime factorisation of 2 is 2. Since it has 2 as it's factor, there is a natural number " n " for which $2^{n}$ will end with zero. This can be said by the uniqueness of the Fundamental Theorem of Arithmetic

Third case : Prediction of the unit's digit for the number with expansion
( 191 ) ¹⁰⁰⁹

We know that the number 1 raised to any power gives 1 as the answer in the unit's place. That is,

$1^{n}$ will always have 1 at it's unit place.

Here ' n ' is any natural number

Hence we can say that,

( 191 ) ¹⁰⁰⁹ will have 1 at it's unit's place. 

Hope my answer helped !!

Cheers !!

Answer 2

Heya!

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1.) Suppose,

n = 0, 1, 2 , 3 , 4

26^0 = 1
26^1 = 26
26^2 = 676
26^3 = 17576
26^4 = 456976

And so on.....

Each time, the last digit us 6 except in the first case , ie - 26^0 = 1.

Thus, 26^n cannot end with 5.

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2.) To end a number with 6, it should have 2 as it's one of the prime factors,

And as 2^n has 2 as it's one of the factors ( 2 × 1)^n
2^n will end with 6 only if it's divisible by 4.
So yes it can end with 6.

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3.)

(191) ^ 1009

We know,

1^n = 1

[ 1 raised to any power always gives 1 ]

So, we conclude, that (191)^1009 will have 1 at its unit place :)

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Hope it helps...!!!