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

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x is a positive integer, and the units digits of both (x+2)^2 and (x-2  [#permalink]

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New post 08 Jan 2019, 00:22
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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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Re: x is a positive integer, and the units digits of both (x+2)^2 and (x-2  [#permalink]

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New post 08 Jan 2019, 01:00
The only number which satisfies is 5, as \((5+2)^2 = 49\) and \((5-2)^2 = 9\)
Then x is 5.

C is the answer.
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Re: x is a positive integer, and the units digits of both (x+2)^2 and (x-2  [#permalink]

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New post 08 Jan 2019, 01:08
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MathRevolution 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 \(x\)?

\(A. 1\)
\(B. 3\)
\(C. 5\)
\(D. 7\)
\(E. 9\)



Back Solving is fine here.

Option C) 5.

we must add the unit digit and square it. Unit digit we will get must be 9.

\((x+2)^2 = (10x5 +2)^2\)= y9........just analyzing unit digit.

\((x-2)^2 =(10m5 - 2)^2\) = n9

In answer choices unit digit of x has been given . add unit digit and square it.

C is the best answer.
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Re: x is a positive integer, and the units digits of both (x+2)^2 and (x-2  [#permalink]

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New post 08 Jan 2019, 01:17
MathRevolution 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 \(x\)?

\(A. 1\)
\(B. 3\)
\(C. 5\)
\(D. 7\)
\(E. 9\)


Which value when substituted \((x+2)^2\) and \((x-2)^2\) will give the value as 9

Plug in from the answer options, start from C, \((5+2)^2\) and \((5-2)^2\)
=> \(49\) and \(9\)

Making the answer as C
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Re: x is a positive integer, and the units digits of both (x+2)^2 and (x-2  [#permalink]

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New post 08 Jan 2019, 01:27
1
MathRevolution 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 \(x\)?

\(A. 1\)
\(B. 3\)
\(C. 5\)
\(D. 7\)
\(E. 9\)



plugin answer choices and solve

we would realize that only option c 5 goes well

IMO C
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Re: x is a positive integer, and the units digits of both (x+2)^2 and (x-2  [#permalink]

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New post 08 Jan 2019, 04:06
1
MathRevolution 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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Re: x is a positive integer, and the units digits of both (x+2)^2 and (x-2  [#permalink]

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New post 10 Jan 2019, 00:29
=>

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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Re: x is a positive integer, and the units digits of both (x+2)^2 and (x-2  [#permalink]

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New post 26 Jan 2019, 19:38
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Squared integers cannot produce every digit from 0-9 in the units digit. Only the units digit of the number being squared affects the units digit of the product e.g. \(123467^2\) will have the same units digit as \(7^2\)

\(0^2=0\)

\(1^2=1\)

\(2^2=4\)

\(3^2=9\)

\(4^2=16\) (but we only care about the units digit 6)

\(5^2=5\)

\(6^2=6\)

\(7^2=9\)

\(8^2=4\)

\(9^2=1\)

So, only squared integers with a 3 unit digit or a 7 unit digit will result in a 9 unit digit. (and 2, 3, 7, 8 aren't produced at all)

3+2=5=7-2
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Re: x is a positive integer, and the units digits of both (x+2)^2 and (x-2   [#permalink] 26 Jan 2019, 19:38
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