Question

find the last digit of 587^1997​

Answer 1

Answer:

For ex

Step-by-step explanation:

ANSWER

(27)

27

=(3

3

)

27

=3

81

Now we know the cyclioity of 3 is 4. Hence, 81/4 leaves a remainder of 1.

Therefore, the last digit will be 3.

Now,

3

81

=3×3

80

=3×(3

2

)

40

=3×9

40

=3×(10−1)

40

=3×(10

40

−

40

C

1

×10

39

−....−

40

C

38

×10

2

−

40

C

39

×10+1)........(i)

For the last two digits, divide the above expression by 100. Each term of the above expression contains 10

2

except 1.

∴3

81

=3×(100λ+1)

=300λ+3

Therefore, the last two digits will be 03

For the last three digits, divide equation (i) by 1000. Each term of the above expression contains 10

3

except −

40

C

39

×10+1

=−400+1

=−399

∴3

81

=3×(1000λ+(−399))

=3000λ−1197

Therefore, the last three digits will basically be the remainder of −1197/1000=803

Therefore, the last three digits are 803

Answer 2

Answer:

Last digit of ${587}^{1997}$ is 7.

Step-by-step explanation:

Given,

${587}^{1997}$

We want to find last digit or unit digit of the given term.

Here base is 587 and power is 1997.

And last digit of 587 is 7.

We know,power of 7 is 1 then unit digit will be 7,if power of 7 is 2 then unit digit will be 9,if power of 7 is 3 then unit digit will be 3 and if power of 7 is 4 then unit digit will be 1.

This cyclicity table always followed by 7.

Here power of 7 is 1997.

We are dividing 1997 by 4 and get 1 as remainder.

So, unit digit of ${547}^{1997}$ is ${7}^{1} = 7$

This is a problem of Algebra.

Some important Algebra formulas:

${(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} \\ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} \\ {(x + y)}^{3} = {x}^{3} + 3 {x}^{2} y + 3x {y}^{2} + {y}^{3} \\ {(x - y)}^{3} = {x}^{3} - 3 {x}^{2} y + 3x {y}^{2} - {y}^{3} \\ {x}^{3} + {y}^{3} = {(x + y)}^{3} - 3xy(x + y) \\ {x}^{3} - {y}^{3} = {(x - y)}^{3} + 3xy(x - y) \\ {x}^{2} - {y}^{2} = (x + y)(x - y) \\ {x}^{2} + {y}^{2} = {(x - y)}^{2} + 2xy \\ {x}^{2} - {y}^{2} = {(x + y)}^{2} - 2xy \\ {x}^{3} - {y}^{3} = (x - y)( {x}^{2} + xy + {y}^{2} ) \\ {x}^{3} + {y}^{3} = (x - + y)( {x}^{2} - xy + {y}^{2} )$

Know more about Algebra,

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