Binomial Theorem used for prediction of last digit of a large number

Binomial Theorem used for prediction of last digit of a large number
Binomial Expression
#math #algebra #iitjeemaths

[nasrullahhusnan]

Last digit of 7¹ is 7
7² is 9
7³ is 3
7⁴ is 1
7⁵ is 7 repeated
As 4 | 256 therefore 17²⁵⁶ is 1.

[firelogik]

I have an alternate solution to this problem that is in my opinion, a little simpler.
if we can represent the power in the form (4n) or (4n+1) or (4n+2) or (4n+3)
then the last digit can be determined using the last digit of the base.

any number ending with 7 raised to the power or 4n will always have one in its unit digit. Hence 17^256 = 17^(4*64) Hence the units digit becomes one.

Here is a simple table showing the pattern I just talked about:


Units digit | (4n+1) | (4n+2) | (4n+3) | (4n+4)

2 | 2 | 4 | 8 | 6
3 | 3 | 9 | 7 | 1

4 | 4 | 6 | 4 | 6
5 | 5 | 5 | 5 | 5
6 | 6 | 6 | 6 | 6
7 | 7 | 9 | 3 | 1
8 | 8 | 4 | 2 | 6
9 | 9 | 1 | 9 | 1

[tai-shingchau]

7x7=49 --> 49x49=....1; i.e. (...7)^4 = ....1, this recurs every ^4. 256 is a multiple of 4, so the last digit of (....7)^4xn is 1