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The four-digit number 486X, where X is the units digit of 48

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by Max@Math Revolution » Wed Oct 31, 2018 11:27 pm

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[Math Revolution GMAT math practice question]

The four-digit number 486X, where X is the units digit of 486X, is a multiple of 36. What is the value of X?

A. 0
B. 2
C. 4
D. 6
E. 8
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by fskilnik@GMATH » Thu Nov 01, 2018 5:35 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

The four-digit number 486X, where X is the units digit of 486X, is a multiple of 36. What is the value of X?

A. 0
B. 2
C. 4
D. 6
E. 8
$$? = X$$
$${{\left\langle {486X} \right\rangle } \over {{2^2} \cdot {3^2}}} = {\mathop{\rm int}} \,\,\,\, \Rightarrow \,\,\,\,\left\{ \matrix{
\,\,\left\langle {486X} \right\rangle \,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,4\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {6X} \right\rangle \,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,4\,\,\,\,\, \Rightarrow \,\,\,\,X \in \left\{ {0,4,8} \right\}\,\,\,\left( * \right) \hfill \cr
\,\,\left\langle {486X} \right\rangle \,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,9\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {18 + X\,\,,\,\,{\rm{hence}}} \right)\,\,X\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,9\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\,? = X = 0 \hfill \cr} \right.$$

The correct answer is therefore (A).


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

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Scott@TargetTestPrep » Fri Nov 02, 2018 4:04 pm
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

The four-digit number 486X, where X is the units digit of 486X, is a multiple of 36. What is the value of X?

A. 0
B. 2
C. 4
D. 6
E. 8
Since 36 = 4 x 9, we see that 486X must be divisible by 4 and 9. From the divisibility rules, we see that the sum of the digits must be a multiple of 9, and the number formed by the last two digits, 6X, must be a multiple of 4.

Since 4 + 8 + 6 = 18, so X can be 0 or 9 if 486X is divisible by 9, or it can be 0, 4 or 8 if 486X is divisible by 4. However, since 486X is divisible by both 9 and 4, X must be 0.

Answer: A

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by Max@Math Revolution » Sun Nov 04, 2018 6:57 pm
=>

Since X is a one-digit integer, 0 ≤ X ≤ 9.
36 = 4*9, so 486X is a multiple of both 4 and 9.
Since 486X is a multiple of 4, 6*10 + X must be a multiple of 4.
Thus, X is one of values 0, 4 and 8.
In addition, since 486X is a multiple of 9, the sum of all of its digits, 4 + 8 + 6 + X = X + 18 is a multiple of 9.
Thus, X must be 0 or 9.

0 is the only one of these digits for which 486X is a multiple of 36.

Therefore, the answer is A.
Answer: A
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