Question

The product of unit’s digit in \[\left( {{7^{95}} - {3^{58}}} \right)\] and \[\left( {{7^{95}} + {3^{58}}} \right)\] :
A.is cube of \[2\]
B.lies between \[6\] and \[10\]
C.is \[6\]
D.lies between \[3\] and \[6\]

Answer 1

Hint: We have to separately find the possible unit digit of each term and then perform the operation to get the final unit digit of the given expression. Find out the exponents which give the unit digit \[1\] and then multiply it to get the unit digit of the particular exponent. The unit digit remains unaffected by the other digits irrespective of any mathematical operation. Operating unit digit of each term will give the unit digit of the final answer.

Complete step by step solution:
We have to find the unit digit of the product of \[\left( {{7^{95}} - {3^{58}}} \right)\] and \[\left( {{7^{95}} + {3^{58}}} \right)\] .
We will get the unit digit of each term separately and then get the result.
Starting with \[{7^{95}}\]
Unit digit of \[{7^{95}} = \] unit digit of \[{\left( {{7^4}} \right)^{23}} \times {7^3}\] as power could be expanded by its factor.
Now Expanding \[{7^4}\] for getting its unit digit
 \[{7^4} = 7 \times 7 \times 7 \times 7 = 2401\]
That is the unit digit is \[1\] and so the exponents of \[{7^4}\] will also have unit digit as \[1\]
So proceeding further
Unit digit of \[{\left( {{7^4}} \right)^{23}} \times {7^3}\] is equal to the unit digit of \[{7^3}\] as the unit digit of the exponents of \[{7^4}\] will also equal to the unit digit of \[{7^4}\] that is \[1\] .
Which is equal to the unit digit of \[343\] as a cube of 7 is 343.
Now, unit digit of \[{3^{58}}\] is equal to the unit digit of \[{\left( {{3^4}} \right)^{14}} \times {3^2}\] .
Now expanding \[{3^4}\] for getting its unit digit
  \[{3^4} = 3 \times 3 \times 3 \times 3 = 81\]
That is, the unit digit is \[1\] and so the exponents of \[{3^4}\] will also have a unit digit as \[1\] .
So the unit digit of \[{\left( {{3^4}} \right)^{14}} \times {3^2}\] is equal to the unit digit of \[{3^2}\] which is equal to \[9\] as the unit digit of the exponents of \[{3^4}\] will also equal to the unit digit of \[{3^4}\] that is \[1\] .
Therefore the unit digit in \[{7^{95}} - {3^{58}}\] is equal to the unit digit in \[343 - 9\] that is the unit digit in \[334 = 4\]

Also, the unit digit in \[{7^{95}} + {3^{58}}\] is equal to the unit digit in \[343 + 9\] that is the unit digit in \[352 = 2\]
So the unit digit of the product will be equal to unit digit of the product of the unit digits of \[{7^{95}} - {3^{58}}\] and \[{7^{95}} + {3^{58}}\] which is equal to \[4 \times 2 = 8\] .
And \[8\] is also equal to the cube of \[2\] as \[2 \times 2 \times 2 = 8\]
Hence option A and B correct.
So, the correct answer is “Option A and B”.

Note: The method by which this question is solved is a part of a mathematical method termed as Mathematical modulo. This method is also used to check divisibility of any number. It is irrelevant to get the whole result just to get the unit digit. So avoid extra unnecessary calculations to get the result.