[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
The four-digit number 486X, where X is the units digit of 48
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- Max@Math Revolution
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$$? = X$$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
$${{\left\langle {486X} \right\rangle } \over {{2^2} \cdot {3^2}}} = {\mathop{\rm int}} \,\,\,\, \Rightarrow \,\,\,\,\left\{ \matrix{
\,\,\left\langle {486X} \right\rangle \,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,4\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {6X} \right\rangle \,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,4\,\,\,\,\, \Rightarrow \,\,\,\,X \in \left\{ {0,4,8} \right\}\,\,\,\left( * \right) \hfill \cr
\,\,\left\langle {486X} \right\rangle \,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,9\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {18 + X\,\,,\,\,{\rm{hence}}} \right)\,\,X\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,9\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\,? = X = 0 \hfill \cr} \right.$$
The correct answer is therefore (A).
This solution follows the notations and rationale taught in the GMATH method.
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Fabio.
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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.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 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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- Max@Math Revolution
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=>
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
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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