What is the units' digit of the following expression (13)^5*(15)^4*(17

This might help:
Of the (13)^5*(15)^4*(17)^5,
(13)^5 = Odd
(17)^5 = Odd
(15)^4 = Unit digit of 5
5*Odd = Unit of 5

Since we need the units digit of (13)^5*(15)^4*(17)^5, we are really looking for the units digit of:

(3)^5 * (5)^4 * (7)^5

The pattern for units digits of powers of 3 is 3-9-7-1, so when 3 is raised to an exponent that is a multiple of 4, the units digit is 1. Thus, 3^4 has units of 1, and so 3^5 has a units digit of 3.

Since a base of 5 raised to an exponent will always end in 5, 5^4 has a units digit of 5.

Next, the pattern for units digits of powers of 7 is 7-9-3-1, so when 7 is raised to an exponent that is a multiple of 4, the units digit is 1. Thus, 7^4 has units of 1 and 7^5 has a units digit of 7.

Now we multiply the three units digits that we have just obtained. Since 3 x 5 x 7 = 105 (with a units digit of 5), the units digit of (3)^5 * (5)^4 * (7)^5 must also be 5, and thus the units digit of (13)^5 * (15)^4 * (17)^5 is 5.

Answer: D

Using Cyclic rule of unit digit we can find the unit digit of any number written in exponent form. Following steps can be of help.
If a number is written as \(k^{x}\)
Divide the power by 4 and check for remainder.
If remainder =1, unit digit of the answer is going to be same as unit digit of 'k'
If remainder =2, unit digit is going to be the unit digit of \(k^{2}\)
If remainder =3, unit digit is going to be the unit digit of \(k^{3}\)
If remainder =0, unit digit is going to be the unit digit of \(k^{4}\)

For example: To find unit digit of \(2017^{35}\)
Divide the power,35, by 4. Check the remainder
\(\frac{35}{4}\)gives remainder 3
Thus \(2017^{35}\)will have last digit same as last digit of \(2017^{3}\)
7*7*7 gives last digit as 3
Thus answer of above example is 3.

Concept: whenever a number is divisible by 5, the units digit must be either:

0 or 5

Since (15)^3 is a factor in the multiplication chain, the result must be a multiple of 5.

Thus, only A or D is a possible answer

Concept 2: in order to have a units digit of 0 (a “trailing zero”) the prime factorization of the product needs to include at least 1 prime factor pair of (2 x 5)

Since all 3 Bases are ODD, the result must be ODD, and can NOT end in a units digit of 0

Answer must be (d) 5

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