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# If n is a positive integer, what is the units digit of n?

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Re: if n is a positive integer, what is the units digit of n? [#permalink]
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if n is a positive integer, what is the units digit of n?

1) the units digit of $$(n+4)^2$$ is 4
2) the units digit of $$(n+3)^2$$ is 1

Given that alone they are not sufficient, combining them we get that:
because $$(n+4)^2$$ has a 4 as unit digit, $$n+4$$ or is 8 or is a two digit integer ending with a 2
because $$(n+3)^2$$ has a 1 as unit digit, $$n+3$$ or is 9 or is a two digit integer ending with a 1

Because the question asks for the unit digit of n, we can conclude that n will have a 8 as unit digit:
X8+4=K2
X8+3=K1
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Re: If n is a positive integer, what is the units digit of n? [#permalink]
Statement #1 ) : the unit digit of (n+4 ) ^ 2 is 4 : we know that the unit digit of 8^2 is 4 and also the unit digit of 12^2 . so n+4 could be 8 and 12

so n+4 = 8 and here n =4 OR n+4= 12 and Here n=8 so Two different integer and two different result . so Data # 1 Is insufficient.

Statement#2) : the unit digit of (n+3)^2 is 1 : we know that 9^2 has 1 in its unit digit and also 11^2 has 1 unit digit so n+3 = 9 OR 11 and n could be 6 OR 8 this Data

has 2 possible value and is ruled out too....

Combined info : we get 8 from 2 info in common so sufficient ... Answer is C
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Re: If n is a positive integer, what is the units digit of n? [#permalink]
Unit digit of n: 0 1 2 3 4 5 6 7 8 9
Statement 1:
Unit digit of n+4: 4 5 6 7 8 9 0 1 2 3
Unit digit of(n+4)^2: 6 5 6 9 4 1 0 1 4 9
Thus, unit digit can be 4 or 8, insufficient

Statement 2:
Unit digit of n+3: 3 4 5 6 7 8 9 0 1 2
Unit digit of(n+3)^2: 9 6 5 6 9 4 1 0 1 4
Thus, unit digit can be 6 or 8, insufficient

Statement 1 & 2: unit digit is 8, sufficient.
Choose C.
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Re: If n is a positive integer, what is the units digit of n? [#permalink]
MartyMurray KarishmaB Is there any way we can solve this without plugging in numbers ?
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Re: If n is a positive integer, what is the units digit of n? [#permalink]
sayan640

You don't necessarily have to plug in numbers. You just have to recognize that when you square a number, there are two ways to end up with a units digit of 4 and two ways to end up with a units digit of 1. For this purpose, it's useful to become familiar with what happens to each of the digits from 0 to 9 when you raise it to higher powers.

Some numbers never change units digit. If a number ends in 0, 1, 5, or 6, then all positive integer powers of that number will end in the same thing. For instance, when you raise 5 to increasing powers, you get 5, 5, 125, 625, etc.

Some numbers flip between two units digits. If a number ends in 4 or 9, then its subsequent powers will go back and forth between two units digits. For 4, those are 4 and 6 (4, 16, 64, 256, etc.). For 9, those are 9 and 1 (9, 81, 729, 6561, etc.).

The remaining numbers (those ending in 2, 3, 7, and 8) follow patterns of 4 digits, and those look like this:
2: 2, 4, 8, 6
3: 3, 9, 7, 1
7: 7, 9, 3, 1
8: 8, 4, 2, 6

We can use these patterns to assess the statements.

1) If we square something and the result ends in 4, then the original number must have ended in 2 or 6. None of the other units digits fit this pattern. (You can verify by squaring all the other single-digit integers and seeing that none of the results end in 4.) If (n+4) has a units digit of 2 or 6, then n must end in 8 or 2. Insufficient.

2) Same idea here. If we square something and the result ends in 1, then the original number must have ended in 1 or 9. If (n+3) has a units digit of 1 or 9, then n must end in 8 or 6. Insufficient.

1&2) We have two different facts about n: it must end in 8 or 2, and it must end in 8 or 6. Since 8 is the only common value, n must end in 8. Sufficient.
Re: If n is a positive integer, what is the units digit of n? [#permalink]
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