Question

Question for genius only ​

Answer 1

Answer:

Step-by-step explanation:

As, 216=6^3 and for all powers of 6 the unit's digit remains fixed at 6, the unit's digit contribution of the first term is 6.

In 6 square and 6^4 also the unit digit is 6.

Similarly as, 25=5^2 and for all powers of 5 the unit's digit remains fixed at 5, the unit's digit contribution of the second term is 5.

Also in the 5 cube and 5^4 the unit digit is 5.

For example, units digit of 172 will be unit's digit of 72=49 as 9.

Applying this concept on the third term, we get,

Unit's digit of 97^7892= unit's digit of 7^7892.

Unit's digit for 7^1=7 is 7,

Unit's digit for 7^2=49 is 9,

Unit's digit for 7^3=343 is 3,

Unit's digit for 7^4=2401 is 1.

So applying this

216^5192 + 25^4317 + 97^ 7892

The unit place is 6 + 5 + 1 = 12

So unit place is 2

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Answer 2

Step-by-step explanation:

You can solve it by two method.

1) cyclicity

In this method we have to observe the pattern of how the unit digit changing.

Here, It's all depends on unit digit there is no role of tens or hundreds place digit.(They are only for making us scare:p)

If we look the first term which is 216 the unit digit is 6

Now, we can use cyclicity,

6¹ = 6

6² = 36

6³ = 1296

6⁴ = 7776

and so on

Here, cylicity of 6 is 1. means,It repeats it's unit value after every one interval.

so, the answer for first term is 6¹ = 6

In second term, which is 25.

5 is its unit place digit.

Now, like earlier whatever the power of 5(expect 0 and fraction, :p) it's always give the 5 as its unit place.

In third term, which is 97

Here, 7 is its ones place digit.

Cyclicity of 7 is

7¹ = 7

7² = 49

7³ = 343

7⁴ = 2401

After this the cycle again repeat so, the cyclicity of 7 is 4.

Now,the cyclicity number i.e. with 4 divide the given power i.e. 7892/4 and the remainder is 0.

If the remainder becomes zero in any case then the unit digit will be the last digit of x^cyclicity number

Here x unit digit of given number.

so, 7⁴ = 2401

Note: If remainder is not zero. Then,x^remainder

(x is again unit digit.)

so, overall answer is 6 + 5 + 1 = 12 so, 2 is required value.

2) Shortcut Method:

$let \: any \: number \: is \: in \: theform \\ of \: {x}^{y}$

Here, x,y ∈ R

1) Find the unit digit of x ;and let it be m.

2) Divide exponent "y" by 4

• If the y is completely divided by 4. i.e remainder is 0.

• a) The unit digit of given expression is 6 if, m = 2,4,6,8

• b) The unit digit of given expression is 1. if, m = 3,7,9

• If y give Non-zero remainder "r". i.e y = 4k + r. Then value of expression is m^r

Note: Here if unit digit is 5 then obvious the unit digit of expression will be 5.