Is the three-digit number n less than 550?

Question

Is the three-digit number n less than 550 ?

(1) The product of the digits in n is 30.
(2) The sum of digits in n is 10.

Bunuel · Accepted Answer

(1) The product of the digits in n is 30 -->  30=1*2*3*5  --> n can be any combination of digits from the following two sets:  {1, 6(=2*3), 5}  or  {2, 3, 5} . n can be lees than 550 (eg. 165, 156, ...) OR more than 550 (eg 615, 651). Not sufficient.
 
(2) The sum of the digits in n is 10 --> Clearly not sufficient. Many combinations are possible 109, 901, ....

(1)+(2) We have the set  {2, 3, 5}  (as only in this set the sum of the digits is 10). From this set we can not form three digit number more than 550 (max=532). Sufficient.

Answer: C.

IanStewart · Answer

Several things here:

- you don't have three equations; you only have two. The expression "100x + 10y + z" is not an equation (there's no equals sign);

- when your equations are not linear -- that is, when unknowns are raised to powers, divided by each other, multiplied together, etc -- counting equations is pretty much never going to tell you anything useful. I can easily generate 50 non-linear equations in 3 unknowns that still cannot be solved. For example, the 5 equations below will all be true as long as z=0; x and y could be anything at all (as long as the denominators are nonzero):

xyz = 0
xz + yz = 0
(x^2)z + z/y = 0
z/(x+y) = 0
z^3 + xz^2 - yz = 0

- even when you have linear equations, if there are restrictions - for example, if your unknowns must be positive integers - you often need much less information than you might expect. For example, if I tell you that a, b, c, d, e, f and g are all positive integers, and that:

a + b + c + d + e + f + g = 7

then I have just one equation in 7 unknowns, but I can solve; a, b, c, d, e, f and g must all be equal to 1.

____

There is a simple rule in algebra when you see two distinct linear equations and two unknowns; then you can always solve for your unknowns. If you have more than 2 unknowns, you have non-linear equations, you have equations which may not be distinct, you have restrictions on your unknowns (for example, if they must be integers), or you have a question which asks for the value of some expression and not some individual unknown, then you cannot easily predict how many equations you will need to have to solve; you normally need to do some work.

Many prep books suggest counting equations and unknowns as a 'shortcut' in GMAT Data Sufficiency - they say that you can't solve if you have fewer equations than unknowns, and you can solve if you have as many equations as unknowns. That's a gross oversimplification of mathematics, and I find that to be among the most misleading advice that prep books give, especially for higher level test takers, who will encounter questions which test the exceptions to these overly simplistic 'rules'. Yes, by counting equations and unknowns, you will get an answer quickly, but if you want to quickly get a wrong answer, you might as well just guess randomly. It is useful to understand the rule for 2 distinct linear equations, but it's equally important to know when you should not be using a counting equations/unknowns 'rule'.

I'd been meaning to do this at some point, and I had some free time today, so I made a (probably not comprehensive) list of questions in the DS section of OG12 where just counting equations and unknowns will lead you to the wrong answer:

Q15 (you have three unknowns, so the rules don't apply as you might expect - sometimes you only need two equations to find one of the three unknowns)
Q52 (asks for the value of an expression, not of an unknown)
Q53 (asks for the value of an expression, not of an unknown)
Q56 (you have three unknowns, so the rules don't apply as you might expect - sometimes you only need two equations to find one of the three unknowns)
Q83 (asks for the value of an expression, not of an unknown, and the equations are non-linear)
Q84 (asks for the value of an expression, not of an unknown)
Q88 (you have three unknowns, so the rules don't apply as you might expect - sometimes you only need two equations to find one of the three unknowns)
Q99 (severe restrictions on the possible values of each unknown)
Q110 (the equation in the stem is non-linear, and the quantities need to be positive integers)
Q123 (the quantities need to be positive integers)
Q137 (asks for the value of an expression, not of an unknown)
Q150 (asks for the value of an expression, not of an unknown)
Q168 (asks for the value of an expression, not of an unknown)

and in the DS section of the green Quant Review book:

Q2 (asks for the value of an expression, not of an unknown)
Q4 (asks for the value of an expression, not of an unknown)
Q23 (the first equation only appears to contain two unknowns; one of them cancels)
Q28 (asks for the value of an expression, not of an unknown, and the equations are non-linear)
Q33 (you have three unknowns, so the rules don't apply as you might expect - sometimes you only need two equations to find one of the three unknowns)
Q37 (asks for the value of an expression, not of an unknown)
Q46 (asks for the value of an expression, not of an unknown, and the equations are non-linear)
Q48 (asks about expressions, not individual unknowns)
Q54 (asks for the value of an expression, not of an unknown)
Q73 (asks for the value of an expression, not of an unknown, and the equations are non-linear)
Q94 (asks for the value of an expression, not of an unknown)
Q100 (asks for the value of an expression, not of an unknown, and the equations are non-linear)
Q106 (asks about expressions, not individual unknowns)
Q108 (equations are not distinct)
Q109 (asks about expressions, not individual unknowns)

from which you can see how frequently GMAT questions are designed to 'trap' people who simply count equations and unknowns without understanding when it is useful to do so and when it is misleading. I listed the question numbers above, since I think it can be instructive to look at a few of these questions together, to see the variety of ways the GMAT will try to trap people who are simply counting unknowns and equations.

Bunuel · Answer

Is the three-digit number n less than 550?

(1) The product of the digits in n is 30 -->  30=1*2*3*5  --> n can be any combination of digits from the following two sets:  {1, 6(=2*3), 5}  or  {2, 3, 5} . n can be lees than 550 (eg. 165, 156, ...) OR more than 550 (eg 615, 651). Not sufficient.
 
(2) The sum of digits in n is 10 --> Clearly not sufficient. Many combinations are possible 109, 901, ....

(1)+(2) We have the set  {2, 3, 5}  (as only in this set the sum of the digits is 10). From this set we can not form three digit number more than 550 (max=532). Sufficient.

Answer: C.

LM · Answer

Excellent Bunuel. I missed the factor "1" when checking the factors of "30". And if you miss this "1" as the factor the answer you end up with is "A", which indeed is wrong.

saxenashobhit · Answer

ha....I also forgot "1" as a factor....thanks for reminding.....

vigneshpandi · Answer

GGRRRRRRR.....Made the same mistake...Forgot to include 1 and chose the answer as A.
I should not make similar mistakes in the real exam.
Thanks BUnnel.

maverick04 · Answer

Isnt there a better way ...?

Any three digit number is 100x + 10y + z

From 1 we have xyz = 30
From 2 we have x+y+z = 10

We have three variables and three equations ..

Without solving, cant we guess that solution to these three equations will give us an answer ... ?

fskilnik · Answer

100 \leqslant \,\,n = \left\langle {ABC} ightangle \,\, \leqslant 999

\left\langle {ABC} ightangle \,\,\mathop < \limits^? \,\,550

\left( 1 ight)\,\,\,A \cdot B \cdot C = 30\,\,\,\left\{ \begin{gathered}
 \,{	ext{Take}}\,\,\left( {A,B,C} ight) = \left( {1,5,6} ight)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{	ext{YES}}} ightangle \hfill \
 \,{	ext{Take}}\,\,\left( {A,B,C} ight) = \left( {5,6,1} ight)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{	ext{NO}}} ightangle \hfill \ 
\end{gathered} ight.

\left( 2 ight)\,\,\,A + B + C = 10\,\,\,\left\{ \begin{gathered}
 \,{	ext{Take}}\,\,\left( {A,B,C} ight) = \left( {1,2,7} ight)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{	ext{YES}}} ightangle \hfill \
 \,{	ext{Take}}\,\,\left( {A,B,C} ight) = \left( {5,5,0} ight)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{	ext{NO}}} ightangle \hfill \ 
\end{gathered} ight.

\left( {1 + 2} ight)\,\,\,\left\langle {ABC} ightangle \,\,\, \geqslant \,\,\,550\,\,\,\, \Rightarrow \,\,\,\left\{ \begin{gathered}
 \,A = 5\,\,,\,\,B \geqslant 5\,\,\,\mathop \Rightarrow \limits^{\left( 2 ight)} \,\,\,B = 5,\,\,C = 0\,\,\, \Rightarrow \,\,\,\left( 1 ight)\,\,{	ext{contradicted}} \hfill \
 \,\,\,{	ext{or}} \hfill \
 \,A \geqslant 6\,\,\,\mathop \Rightarrow \limits_{\left( 1 ight)}^{\left( * ight)} \,\,\,A = 6\,\,\,\mathop \Rightarrow \limits^{\left( 1 ight)} \,\,\,B \cdot C = 5\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} ight)} \,\,\,A + B + C = 12\,\,\, \Rightarrow \,\,\,\left( 2 ight)\,\,{	ext{contradicted}} \hfill \ 
\end{gathered} ight.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{	ext{YES}}} ightangle

\left( * ight)\,\,\,\left( 1 ight)\,\,\, \Rightarrow \,\,\,A,B,C\,\,{	ext{are}}\,\,\left( {{	ext{positive}}} ight)\,\,{	ext{divisors}}\,\,{	ext{of}}\,\,30\,\,\, \Rightarrow \,\,\,A 
otin \left\{ {7,8,9} ight\}

\left( {**} ight)\,\,\,B \cdot C = 5\,\,\, \Rightarrow \,\,\,B,C\,\,{	ext{is}}\,\,{	ext{a}}\,\,{	ext{pair}}\,\,{	ext{of}}\,\,\left( {{	ext{positive}}} ight)\,\,{	ext{divisors}}\,\,{	ext{of}}\,\,5\,\,\, \Rightarrow \,\,\,\left\{ \begin{gathered}
 \,\left( {B,C} ight) = \left( {1,5} ight) \hfill \
 \,\,{	ext{or}} \hfill \
 \,\left( {B,C} ight) = \left( {5,1} ight) \hfill \ 
\end{gathered} ight.

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

Regards, 
Fabio.

PabloOli · Answer

wait,

1) The question says the product of the digits in n is 30.

2) n is a three-digit number

3) Those three digits are 2,3 and 5.

1 should not be included, only 2,3, and 5. The question does not say "the product of the factors of n" it says the products of the digits in n,

Without 1, the correct answer is A.

Bunuel · Answer

The correct answer is C, not A. For (1): n can be lees than 550 (eg. 165, 156, ...) OR more than 550 (eg 615, 651). Not sufficient.

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Is the three-digit number n less than 550?

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