Problem Solving

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

Answer: [spoiler]____(C)__[/spoiler]

fskilnik@GMATH 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

Answer: [spoiler]____(C)__[/spoiler]

Call X the number to be divided by 5. So, X = 5K + 1 represents this number with remainder 1.

Examining this shows that for K = 1, 2, 3...etc. a repeating pattern of 6,1,6,1..., so the number X will have either a 1 or a 6 as the units digit.

4758 units less than this number X means subtracting the units digit 8 from a units digit of either 1 or 6, which in turn will leave a units digit of either 3 or 8 for the resulting number.

So the product of 118A and 25847 will have a units digit of either 3 or 8. This means that A x 7 will have a units digit of either 3 or 8.

Trying possible digits for A from 0 to 9 shows that only 4 and 9 yield a 3 or an 8, answer C, 2

fskilnik@GMATH 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

$$\left\langle {118A} \right\rangle \cdot 25847 + 4758 = 5M + 1\,\,\,,\,\,\,M \ge 1\,\,{\mathop{\rm int}} $$
$$?\,\,\,:\,\,\,\# \,\,A\,\,{\rm{possibilities}}$$
$$\left[ N \right] = units\,\,digit\,\,of\,\,N\,\,\,\,\,\left( * \right)$$
$$\left\langle {118A} \right\rangle \cdot 25847 + 4757 = 5M\,\,\,\,\,{\rm{AND}}\,\,\,\,\,\left( * \right)\,\,\,\left\{ \matrix{
\,\,\left[ {\left\langle {118A} \right\rangle \cdot 25847} \right] = \left[ {7 \cdot A} \right] \hfill \cr
\,\,\left[ {4757} \right] = 7 \hfill \cr
\,\,\left[ {5M} \right] = 0 \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left[ {7 \cdot A + 7} \right]\,\,\,{\rm{equals}}\,\,\,0\,\,{\rm{or}}\,\,5$$
$$ \Rightarrow \,\,\,\,\,7 \cdot \left( {A + 1} \right)\,\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,5\,\,\,\,\mathop \Rightarrow \limits^{GCD\left( {7,5} \right)\,\, = \,\,1} \,\,\,\,\,\left[ {A + 1} \right]\,\,\,{\rm{equals}}\,\,\,0\,\,{\rm{or}}\,\,5\,\,\,\,\, \Rightarrow \,\,\,A\,\,\,{\rm{equals}}\,\,{\rm{4}}\,\,{\rm{or}}\,\,{\rm{9}}$$


The correct answer is therefore (C).


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

Regards,
Fabio.