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x is a single-digit positive integer. If the units digit of x^2 and

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mangamma
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x is a single-digit positive integer. If the units digit of x^2 and [#permalink] New post   13 May 2019, 08:59
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x is a single-digit positive integer. If the units digit of x^2 and the units digit of (x+2)^3 is same. How many values of x are possible?


A. 2
B. 3
C. 4
D. 7
E. 9
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C
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MofeBhatia
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x is a single-digit positive integer. If the units digit of x^2 and [#permalink] New post   13 May 2019, 10:57
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I didn't find any easy or faster method to solve this.
2,4,7,9 satisfy the equation unitdigit(x2) = unitdigit(x+2)3
x^2 (x+2)^3
2 4 4^3 = 64
4 16 6^3 = 216
7 49 9^3 = 729 = 81 * 9 unit digit = 9
9 81 11^3 = unitdigit = 1

Please let me know if you some other method
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Re: x is a single-digit positive integer. If the units digit of x^2 and [#permalink] New post   14 May 2019, 19:24
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mangamma wrote:
x is a single-digit positive integer. If the units digit of x^2 and the units digit of (x+2)^3 is same. How many values of x are possible?


A. 2
B. 3
C. 4
D. 7
E. 9


Let’s test each possible units digit of x^2 and (x + 2)^3 (note: in the calculations below, we only care about the units digit of x^2 and (x + 2)^3):

1^2 = 1 and 3^3 = 7

2^2 = 4 and 4^3 = 4 → Yes!

3^2 = 9 and 5^3 = 5

4^2 = 6 and 6^3 = 6 → Yes!

5^2 = 5 and 7^3 =3

6^2 =6 and 8^3 = 2

7^2 = 9 and 9^3 =9 → Yes!

8^2 = 4 and 10^3 = 0

9^2 = 1 and 11^3 = 1 → Yes

Answer: C
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KarishmaB
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Re: x is a single-digit positive integer. If the units digit of x^2 and [#permalink] New post   23 Apr 2021, 01:54
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mangamma wrote:
x is a single-digit positive integer. If the units digit of x^2 and the units digit of (x+2)^3 is same. How many values of x are possible?


A. 2
B. 3
C. 4
D. 7
E. 9


You will need to check for digits from 1 to 9 though you can eliminate some pairs right away.

x = 1, (x + 2) = 3
1^1 = 1, 3^3 = 27

x = 2, (x + 2) = 4
2^2 = 4, 4^3 = 64 - Hit

x = 3, (x + 2) = 5
No common units digits

x = 4, (x + 2) = 6
4^2 = 16, 6^3 = 216 - Hit

x = 5, (x+2) = 7
No common units digits

x = 6, (x+2) = 8
6^2 = 36, 8^3 = 512

x = 7, (x + 2) = 9
7^2 = 49, 9^3 = 729 - Hit

x = 8, (x+2) = 10
No common units digits

x = 9, (x+2) = 11
9^2 = 81, 11^3 = 1331 (or at least we know that it ends in 1) - Hit

Answer (C)
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Re: x is a single-digit positive integer. If the units digit of x^2 and [#permalink]
23 Apr 2021, 01:54
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