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# The product of the positive 4-digit integer 118A and 25847 is 4758 uni

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GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
The product of the positive 4-digit integer 118A and 25847 is 4758 uni  [#permalink]

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07 Feb 2019, 11:49
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Difficulty:

45% (medium)

Question Stats:

69% (02:40) correct 31% (02:35) wrong based on 26 sessions

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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

_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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Re: The product of the positive 4-digit integer 118A and 25847 is 4758 uni  [#permalink]

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07 Feb 2019, 22:40
fskilnik wrote:
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

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
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GMATH Teacher
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The product of the positive 4-digit integer 118A and 25847 is 4758 uni  [#permalink]

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08 Feb 2019, 06:30
fskilnik wrote:
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

Hi, chetan2u ! Thanks for joining and for your nice contribution!

Let me offer our "official solution":

$$\left\langle {118A} \right\rangle \cdot 25847 + 4758 = 5M + 1\,\,\,,\,\,\,M \ge 1\,\,{\mathop{\rm int}}$$

$$?\,\,\,:\,\,\,\# \,\,A\,\,{\rm{possibilities}}$$

$$\left[ N \right] = units\,\,digit\,\,of\,\,N\,\,\,\,\,\left( * \right)$$

$$\left\langle {118A} \right\rangle \cdot 25847 + 4757 = 5M\,\,\,\,\,{\rm{AND}}\,\,\,\,\,\left( * \right)\,\,\,\left\{ \matrix{ \,\,\left[ {\left\langle {118A} \right\rangle \cdot 25847} \right] = \left[ {7 \cdot A} \right] \hfill \cr \,\,\left[ {4757} \right] = 7 \hfill \cr \,\,\left[ {5M} \right] = 0 \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left[ {7 \cdot A + 7} \right]\,\,\,{\rm{equals}}\,\,\,0\,\,{\rm{or}}\,\,5$$

$$\Rightarrow \,\,\,\,\,7 \cdot \left( {A + 1} \right)\,\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,5\,\,\,\,\mathop \Rightarrow \limits^{GCD\left( {7,5} \right)\,\, = \,\,1} \,\,\,\,\,\left[ {A + 1} \right]\,\,\,{\rm{equals}}\,\,\,0\,\,{\rm{or}}\,\,5\,\,\,\,\, \Rightarrow \,\,\,A\,\,\,{\rm{equals}}\,\,{\rm{4}}\,\,{\rm{or}}\,\,{\rm{9}}$$

The correct answer is therefore (C).

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
The product of the positive 4-digit integer 118A and 25847 is 4758 uni   [#permalink] 08 Feb 2019, 06:30
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