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07 Feb 2019, 11:49
Kudos
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
70% (02:23) correct
30%
(02:16)
wrong
based on 231
sessions
History
Date
Time
Result
Not Attempted Yet
GMATH practice exercise (Quant Class 16)
The product of the positive 4-digit integer 118A and 25847 is 4758 units less than a number that leaves remainder 1 when divided by 5. How many values are possible for the digit A?
(A) None
(B) Only 1
(C) Only 2
(D) Only 3
(E) More than 3
The product of the positive 4-digit integer 118A and 25847 is 4758 units less than a number that leaves remainder 1 when divided by 5. How many values are possible for the digit A?
(A) None
(B) Only 1
(C) Only 2
(D) Only 3
(E) More than 3
07 Feb 2019, 22:40
Kudos
Bookmarks
fskilnik
Let us work on the unit's digit..
For remainder to be 1 when divided by 5, the units digit should be either 1 or 1+5=6..
two cases..
1) units digit as 1..
118A*258477=xyz1-4758..........A*7=s1-8=t3....
Multiplication of 7 by 9 gives a units digit 3..
SO A can be 9..
2) units digit as 6..
118A*258477=xyz6-4758..........A*7=s6-8=t8....
Multiplication of 7 by 4 gives a units digit 8..
SO A can be 4..
Thus 2 values..
C
fskilnik
Hi, chetan2u ! Thanks for joining and for your nice contribution!
Let me offer our "official solution":
\(\left\langle {118A} \right\rangle \cdot 25847 + 4758 = 5M + 1\,\,\,,\,\,\,M \ge 1\,\,{\mathop{\rm int}}\)
\(?\,\,\,:\,\,\,\# \,\,A\,\,{\rm{possibilities}}\)
\(\left[ N \right] = units\,\,digit\,\,of\,\,N\,\,\,\,\,\left( * \right)\)
\(\left\langle {118A} \right\rangle \cdot 25847 + 4757 = 5M\,\,\,\,\,{\rm{AND}}\,\,\,\,\,\left( * \right)\,\,\,\left\{ \matrix{\\
\,\,\left[ {\left\langle {118A} \right\rangle \cdot 25847} \right] = \left[ {7 \cdot A} \right] \hfill \cr \\
\,\,\left[ {4757} \right] = 7 \hfill \cr \\
\,\,\left[ {5M} \right] = 0\,\,{\rm{or}}\,\,5 \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left[ {7 \cdot A + 7} \right]\,\,\,{\rm{equals}}\,\,\,0\,\,{\rm{or}}\,\,5\)
\(\Rightarrow \,\,\,\,\,7 \cdot \left( {A + 1} \right)\,\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,5\,\,\,\,\mathop \Rightarrow \limits^{GCD\left( {7,5} \right)\,\, = \,\,1} \,\,\,\,\,\left[ {A + 1} \right]\,\,\,{\rm{equals}}\,\,\,0\,\,{\rm{or}}\,\,5\,\,\,\,\, \Rightarrow \,\,\,A\,\,\,{\rm{equals}}\,\,{\rm{4}}\,\,{\rm{or}}\,\,{\rm{9}}\)
The correct answer is therefore (C).
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
