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Comment on Last Digit of a Large Product
When I apply the method that
Because:
49 to the power of 18, is 9 to the power of 18, has a pattern cycle of 2. Which means that 18/2=9. So the digit number of 49 to the power of 18 has a units digit of the first number in that pattern. Which is 9.
The same applies for 13 to the power of 36. 3 has a pattern cycle of 4. 36/4=8. Which means the units digit of 13 to the power of 36 is equal to the first number in that cycle, which is 3.
So 9x3=27. With a unit digit of 7. Can someone explain why this is different then the above stated question?
Your calculations are a bit
Your calculations are a bit off.
49^18 has units digit 1, and 13^36 has units digit 1
To see why, let's look for a pattern:
49^1 = 49
49^2 = ??1
49^3 = ??9
49^4 = ??1
We can see that the units digit is 9 when the exponent is ODD, the units digit is 1 when the exponent is EVEN.
13^1 = 13
13^2 = ??9
13^3 = ??7
13^4 = ??1
13^5 = ??3
Here, the units digit is 1 when the exponent is divisible by 4. Since 36 is divisible by 4, we know that 13^36 has units 1.
Here's a similar video: https://www.gmatprepnow.com/module/gmat-powers-and-roots/video/1032
And here's a useful article: https://www.gmatprepnow.com/articles/units-digits-big-powers
Thanks for the clarity. I was
Exactly what i did. Now
I suggest that you read the
I suggest that you read the above article.
If you still have questions, please let me know.
Cheers,
Brent
I did this a much longer way,
I think that's a perfectly
I think that's a perfectly acceptable approach. The truth of the matter is that this question type is already time consuming. Your solution might be SLIGHTLY longer, but only by a few seconds.
@YVONNEGMATPREP2017
you can also use x^0 = 1 to start your cycles
makes it much easier
Sometimes yes and oftentimes
Sometimes yes and oftentimes no.
If we apply it to EVEN bases, that approach can cause confusion.
For example:
4^0 = 1 (units digit = 1)
4^1 = 4 (units digit = 4)
4^2 = 16 (units digit = 6)
4^3 = 64 (units digit = 4)
4^4 = 256 (units digit = 6)
The pattern is 4-6-4-6-4-6-...., so we ignore the 4^0 value.
If I were you, I wouldn't look at x^0 when determining cycles.
Cheers,
Brent
I thought I understood the
Let's examine the units digit
Let's examine the units digit of various powers of 13:
13^1 = 13
13^2 = 169
13^3 = ---7 (we need only focus on the units digit)
13^4 = ---1
13^5 = ---3
13^6 = ---9
13^7 = ---7
13^8 = ---1
etc
Notice that the units digits have a pattern: 3, 9, 7, 1, 3, 9, 7, 1, ...
The cycle = 4
Notice that, if the exponent is a MULTIPLE of 4, then the units digit is 1 (13^4 = ---1, 13^8 = ---1, 13^12 = ---1, etc)
Since 36 is a multiple of 4, we know that 13^36 = ---1
FOLLOW-UP QUESTION: What's the units digit 13^302?
Since 300 is a multiple of 4, we know that 13^300 = ---1
From here, we continue the pattern:
13^301 = ---3
13^302 = ---9
So, the units digit 13^302 is 9
Here's an article that discusses this strategy: https://www.gmatprepnow.com/articles/units-digits-big-powers
Hey Brent,
Thank you for the video.
I do this problem with a slightly different approaches.
Since 13^36 and 3 has 4 cycles, I divide 36 into 3, which gives me no remainder.
When there is no remainder, does it always equal to 1?
For example, if if it's 13^35, the remainder will be 2, and 3^2= 9, so the digit is 9.
But if remainder is 0, will the unit digit always equal to 1?
That strategy work to work
That strategy work to work when the base has 3 as its units digit, but it won't work for other powers.
Here's why it works when the base has 3 as its units digit:
13^1 = 13
13^2 = 169
13^3 = ---7 (we need only focus on the units digit)
13^4 = ---1
13^5 = ---3
13^6 = ---9
13^7 = ---7
13^8 = ---1
etc
Notice that, when the exponent is divisible by 4, the units digit is 1.
So, for subsequent powers, we simply multiply 1 by successive 3's.
However, for most other bases, the units digit is NOT 1 when the exponent is divisible by 4.
Cheers,
Brent