Units digit cycle as follows for 3:
3 9 7 1 3 9 7 1
From smt II if u know the units digit of x is 3 then u know the units digit of x^2 will be 9.
Hence SUFF
Stmt I
The number could have units digit of 1 or 3 SO X^2 units digit could be 1 or 9
INSUFF
B
mgmat 4b V.Good Qs
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Great solution - just wanted to point out that there are other possibilities here: if the units digit of x^4 is 1, then the units digit of x could be 1, 3, 7 or 9.cramya wrote: Stmt I
The number could have units digit of 1 or 3 SO X^2 units digit could be 1 or 9
INSUFF
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this might be a no-brainer one for u guys, but
(2) The units digit of x is 3.
are we assuming that x is a single unit number? or a double digit number?
I mean it just says the ''units'' digit is 3... it can mean the number is 3,13,23,43, etc??
(2) The units digit of x is 3.
are we assuming that x is a single unit number? or a double digit number?
I mean it just says the ''units'' digit is 3... it can mean the number is 3,13,23,43, etc??
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Morgoth
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vkb16 wrote:this might be a no-brainer one for u guys, but
(2) The units digit of x is 3.
are we assuming that x is a single unit number? or a double digit number?
I mean it just says the ''units'' digit is 3... it can mean the number is 3,13,23,43, etc??
It doesnt matter weather you take single digit or double digit number, the units digit will always be same.
You can try for all those numbers, the resulting units digit will always be 1
3^4 = 81
13^4 = 28561
so on and so forth.
Hence for all the numbers ending in 3 the units digit will always be 1 if and only if that number is raised to the power of 4,
Similarly, if the number with units digit of 3 raised to the power of 2 the resulting number's units digit will always be 9.
hope this helps.
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lilu
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I have another question:
Are there any other numbers 1-9 that besides 3 that have similar "special" properties? (I mean the way the units repeat when raising to powers in numbers that end with 3)
Thank you!
Liliya
Are there any other numbers 1-9 that besides 3 that have similar "special" properties? (I mean the way the units repeat when raising to powers in numbers that end with 3)
Thank you!
Liliya
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Stuart@KaplanGMAT
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All of them!lilu wrote:I have another question:
Are there any other numbers 1-9 that besides 3 that have similar "special" properties? (I mean the way the units repeat when raising to powers in numbers that end with 3)
Thank you!
Liliya
Only focusing on the units digit:
1^n always ends in 1
2^n follows the pattern 2, 4, 8, 6, ...
3^n follows the pattern 3, 9, 7, 1, ...
4^n follows the patter 4, 6, ...
5^n always ends in 5
6^n always ends in 6
7^n follows the pattern 7, 9, 3, 1, ...
8^n follows the pattern 8, 4, 2, 6, ...
9^n follows the pattern 9, 1, ...
0^n always ends in 0
Last edited by Stuart@KaplanGMAT on Thu Mar 19, 2009 9:06 pm, edited 1 time in total.
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Jose Ferreira
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Hi Liliya,
4 follows the pattern 4, 6, 4, 6
While memorizing these patterns could be helpful, it is more important to understand HOW we arrive at these patterns, so that you have the tools you need to answer the question even if you forget one of the patterns on test day.
The most important point is that when we multiply any two numbers together, all we need to know in order to find the ones digit of the product is the ones digit of each of the two numbers.
Let's take the example of 7.
7^1 = 7.
7^2 = 49, so the ones digit is 9.
To find the ones digit of 7^3, we do NOT need to multiply 7 * 49. Instead, we can ignore everything but the ones digit, and simply multiply 7 * 9 = 63. Thus, the ones digit of 7^3 is 3.
Similarly, to find the ones digit of 7^4, we do NOT need to multiply 7 * 7 * 7 * 7, or even to multiply 63 (the previous result) by 7. Instead, we can ignore everything but the ones digit, and simply multiply 7 * 3 = 21. Thus, the ones digit of 7^4 is 1.
We can continue like this.
7^5 has a ones digit of 7 * 1 = 7.
7^6 has a ones digit of 7 * 7 = 49, so 9.
7^7 has a ones digit of 7 * 9 = 63, so 3.
7^8 has a ones digit of 7 * 3 = 21, so 1.
This gives us the pattern 7, 9, 3, 1, 7, 9, 3, 1.
4 follows the pattern 4, 6, 4, 6
While memorizing these patterns could be helpful, it is more important to understand HOW we arrive at these patterns, so that you have the tools you need to answer the question even if you forget one of the patterns on test day.
The most important point is that when we multiply any two numbers together, all we need to know in order to find the ones digit of the product is the ones digit of each of the two numbers.
Let's take the example of 7.
7^1 = 7.
7^2 = 49, so the ones digit is 9.
To find the ones digit of 7^3, we do NOT need to multiply 7 * 49. Instead, we can ignore everything but the ones digit, and simply multiply 7 * 9 = 63. Thus, the ones digit of 7^3 is 3.
Similarly, to find the ones digit of 7^4, we do NOT need to multiply 7 * 7 * 7 * 7, or even to multiply 63 (the previous result) by 7. Instead, we can ignore everything but the ones digit, and simply multiply 7 * 3 = 21. Thus, the ones digit of 7^4 is 1.
We can continue like this.
7^5 has a ones digit of 7 * 1 = 7.
7^6 has a ones digit of 7 * 7 = 49, so 9.
7^7 has a ones digit of 7 * 9 = 63, so 3.
7^8 has a ones digit of 7 * 3 = 21, so 1.
This gives us the pattern 7, 9, 3, 1, 7, 9, 3, 1.
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Sorry, I was traumatized by 4 as a child!lilu wrote:Thank you so much, Stuart!
I'm making a flash card!
and what about 4?
Modified my post, no clue why I left 4 out
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