problem

Yes, but actually I can't simplify the other portion. Can you please explain it.

When integers are multiplied, the only digits that affect the units digit of a product are the units digits of the numbers being multiplied.

For instance, the units digit of 9 x 9 is the same as the units digit of 209 x 139. The units digits of both are 1.

So in this case, the only digits that matter are the 7 from 17 and the 3 from 1973.

So essentially the problem becomes this. What is the units digit of (7³)�, or 7¹², and what is the units digit of 3^3², or 3�?

The other thing you need to realize to do this problem is that when an integer is raised to consecutive powers, the resulting units digits repeat in a cyclical pattern.

So to get the answer here, we need the cyclical pattern for 7 and the cyclical pattern for 3.

7¹ = 7 (units digit 7)
7² = 49 (units digit 9)
7³ = _ _ 3 (units digit 3 - I am not going to bother multiplying out 7 x 49 since I only need the units digit. Since 7 x 9 = 63, I know the units digit of 7 x 49 is 3.)
7� = _ _ 1 units digit of 1. (Since 7 x 3 = 21 I know that the units digit of 7 x _ _ 3 is 1.)

Now that I am back to 1 as the units digit, when I multiply by 7 again, I will get a units digit of 7 and the pattern will repeat.

7 9 3 1 7 9 3 1 7 9 3 1

So 7¹² has a units digit of 1.

The pattern for 3 is the following.

3¹ = 3 (units digit 3)
3² = 9 (units digit 9)
3³ = 27 (units digit 7)
3� = 81 (units digit 1)

Then, since the last digit of 81 is 1, the pattern starts over, and so the pattern for 3 is the following.

3 9 7 1 3 9 7 1 3 9 7 1

Now since the units digit of 17¹² is 1 and the units digit of 1973� is 3, I was tempted to subtract 3 from 11 and get 8.

Then I figured out that 1973� > 17¹². So the answer is going to be a negative number and we have to reverse the subtraction to get the last digit.

So 3 - 1 = 2 and the correct answer is B.

where is the qus? Can u pl pl repost?

Sandip, the question is at the top.

Ya.got it sir.

Hi Marty,
Got the ans 8 at first instance.later saw ur post and realised where i went wrong.Can u pl suggest how can i avoid such silly mistakes.thanks.

Got the ans 8 at first instance.later saw ur post and realised where i went wrong.Can u pl suggest how can i avoid such silly mistakes.thanks.

There's no magic pill that will allow you to avoid these kinds of mistakes, but it is worth internalizing the idea, as Marty explained earlier, that when we're trying to determine the units digit of the result of the difference of two numbers, we cannot look at the units digits in isolation. I find it helpful to remind myself of this principle by using simple numbers. If x has a units digit of '6,' and y has a units digit of '2,' for example, then the units digit of x - y isn't necessarily 6-2 = 4, because if x = 16 and y = 22, then x - y = -6. Generally speaking, using simple numbers to confirm our intuition is a helpful strategy when dealing with larger values.

sandipgumtya wrote:Got the ans 8 at first instance.later saw ur post and realised where i went wrong.Can u pl suggest how can i avoid such silly mistakes.thanks.

For me much of what helps with avoiding such errors is being aware of the little uneasiness I feel at times when looking at a question.

In this case, I was ready to do the same thing you did, but there was some little uneasiness I was feeling; on some level I was wondering whether the second number were greater than the first.

So one thing you can do is to seek to be aware of what's going through your mind as you work on answering a question and to be sure to pay attention to any questions you have about what you are doing.

Another thing that helps is, as Brent mentioned, being aware of the types of things you do. Then you can seek to catch yourself in the act, and not do them, before you choose answers.

In my case I had a tendency to make silly calculation errors, doing things like adding 26 and 26 and getting 56. I learned to say to myself, "There you go. DOOOOON'T do that." before I did it.

Overall, you can work to develop an accuracy mindset. There is a whole psychology to doing things accurately and precisely, and even as I am typing this I am noticing the typing errors I have to correct as I go along. Why am I making those totally avoidable errors? In order to type more accurately the first time, I have to develop more of an accuracy mindset. The more one develops an accuracy mindset, the more one will be accurate and precise in everything one does.