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x is a positive integer, and the units digits of both (x+2)^

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x is a positive integer, and the units digits of both (x+2)^

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[Math Revolution GMAT math practice question]

x is a positive integer, and the units digits of both (x+2)^2 and (x-2)^2 are 9. What is the units digit of x?

A. 1
B. 3
C. 5
D. 7
E. 9

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Max@Math Revolution wrote:
[Math Revolution GMAT math practice question]

x is a positive integer, and the units digits of both (x+2)^2 and (x-2)^2 are 9. What is the units digit of x?

A. 1
B. 3
C. 5
D. 7
E. 9
$$x \ge 1\,\,{\mathop{\rm int}} $$
$$? = \left\langle x \right\rangle \,\,\,\,\,\left( {{\rm{units}}\,\,{\rm{digit}}\,\,{\rm{of}}\,\,x} \right)$$
$$\left. \matrix{
\left\langle {{{\left( {x + 2} \right)}^2}} \right\rangle = 9\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {x + 2} \right\rangle = 3\,\,\,\,{\rm{or}}\,\,\,\,\,\,\left\langle {x + 2} \right\rangle = 7\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\langle x \right\rangle = 1\,\,\,\,{\rm{or}}\,\,\,\,\,\,\left\langle x \right\rangle = 5 \hfill \cr
\left\langle {{{\left( {x - 2} \right)}^2}} \right\rangle = 9\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {x - 2} \right\rangle = 3\,\,\,\,{\rm{or}}\,\,\,\,\,\,\left\langle {x - 2} \right\rangle = 7\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\langle x \right\rangle = 5\,\,\,\,{\rm{or}}\,\,\,\,\,\,\left\langle x \right\rangle = 9\,\,\,\, \hfill \cr} \right\}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 5$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

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Max@Math Revolution wrote:
x is a positive integer, and the units digits of both (x+2)² and (x-2)² are 9. What is the units digit of x?

A. 1
B. 3
C. 5
D. 7
E. 9
Another approach is to test the answer choices

A. If the units digit of x is 1, is it the case that units digits of both (x+2)² and (x-2)² is 9?
Well, (x+2)² = (1+2)² = 9. Works!
To test (x-2)², let's let x = 11. We get: (11-2)² = 9² = 81. NO GOOD
ELIMINATE A

B. If the units digit of x is 3, is it the case that units digits of both (x+2)² and (x-2)² is 9?
(x+2)² = (3+2)² = 25. NO GOOD
ELIMINATE B

C. If the units digit of x is 5, is it the case that units digits of both (x+2)² and (x-2)² is 9?
(x+2)² = (5+2)² = 49. Works!
(x-2)² = (5-2)² = 9. Works!

Answer: C

Cheers,
Brent

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=>

Since x is a positive integer and the units digits of both (x+2)^2 and (x-2)^2 are 9, the units digits of x+2 and x-2 must be 7 and 3, respectively.
Thus, the units digit of x is 5.

Therefore, the answer is C.
Answer: C

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Unlimited Access to over 120 free video lessons-try it yourself
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