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GMAT Club Olympics 2024 (Day 9): A student was given an equation

1 2 3 4

andreagonzalez2k

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18 Jul 2024, 18:30

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x, y, and z are different digits
y > x

x can not be 0 (z would be 0 too)
x can not be 1 (y and z would be equal)
x can not be 3 o greater (minimun value would be 3*4=12 and 12 is not one digit number)

x only can be 2 and the only valid combinations are:
2*3=6
2*4=8

minimum x=2
maximum x=2

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18 Jul 2024, 19:00

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Since all are single digit numbers, we can only have x=2 which will satisfy the conditions
Min value of x, y and z =2*3 =6
Max value of x,y and z = 2*4 =8

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Minimum X: 1
Maximum X: 6

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Bunuel

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Screenshot_20240719_081813_Samsung Notes.jpg [ 139.23 KiB | Viewed 934 times ]

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The rules we must follow are:
x*y = z, where x,y and z are distinct digits, these means that their values can be from 0 to 9, inclusive
y>X

Since they are all different digits, x can't be 1, because this will lead to y = z

The only possible answe for X is 2.

If x >=3, this will lead to y>=4, and in this combination z will have two digits, wich cannot be possible

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18 Jul 2024, 20:41

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Given $ x * y = z$, where x, y, and z are different digits
and $y > x$

The least possible value of x
x cannot be 0, because then z = 0 (we want different digits)
x cannot be 1, because then z = y (we want different digits)
x can be 2, 2*3 = 6 (we have different digits and y>x)
Minimum x = 2

The maximum possible value of x
x cannot be 9, because there is no y > x
x cannot be 8, 7, 6, 5, 4, or 3 because if y>x then z is a digit not a 2-digit number

$8 * 9 = 72$
$7 * 8 = 56$
$6 * 7 = 42$
$5 * 6 = 30$
$4 * 5 = 20$
$3 * 4 = 12$
$2 * 4 = 8$
Maximum x = 2

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Bunuel

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Since x, y and z are distinct digits and y>x, we’ll start by taking x = 0, which would make the product 0*y = 0, but x ≠ z. Similarly putting x= 1, gives us the equation y*1 = z, again, y = z not possible.

Thus for Minimum x x = 2

We continue by trying x = 3, naturally followed by trying y = 4, (y>x) which gives us z = 12, which is not possible z being a single digit.

There for Maximum x x = 2

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I think it says that x,y,z are different digits because each of these must be a single-digit number (below 10).

If the numbers must be different, we can obviously get rid of 1 for x, because then y=z.
As for 2, it works: $2*3=6$ (for instance).

Let's look for max value now. Given z is a product, it must be non-prime, so all possible options are: 4,6,8,9
$4 = 2*2$
$6 = 2*3$
$8 = 2*4$
$9 = 3*3$
From what we see, the only two options with different digits have $x = 2$
This means that the maximum value for x is aso 2.

Both minimum and maximum values for x = 2.

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Consecutive integers −2 to 5 are ->>>>(-2,-1,0,1,2,3,4,5)
as Repetation allowed

Smallest possible value of the product of the 4 integers for Smallest value-->> 1*1*1*-1=-1

Largest negative value of the product of the 4 integers for Largest negative value-->> 5*5*5*-2=-250

Ans-AE.

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x=y*z all diff digits

So digits can be 0-9 inclusive

Min x--

If X=2 So y=3

2*3=6 Possible

X=3 So y=4

3*4=12 Not possible so

Min value of x=2

Max x--

Again if we choose x=2, y=3

then 2*3=6 possible

if x=3 y=4

3*4=12 Not possible so

max value of x=2

Ans
Min value of x=2
max value of x=2

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Quote:

Since, x, y and z are different digits.

Let's start with minimum value of x => Taking x = 1, 1*y=z, y=z, this is wrong since all three digits are different.
Taking x=2, y has to be atleast 3, since y>x, z => 2*3 = 6

Hence, minimum value of x => 2

Let's try taking a value greater than 2 for x => 3, let's take y=4, z = 3*4 = 12, 12 is not a digit. Hence, the max value of x is also limited to 2.

Minimum x => 2
Maximum x => 2

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This questions only has 1 solution of x value as 2.
2*3=6
The other possibilities are 2*4=8 but 4 here is "y" since y>x.
x can't be 0 or 1 since z cannot be equal to x or y.
x can't be 3 since y has to be bigger so, the least y would be 4 and 12 is not a digit.

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Question stem tells us that,
1. Equation is $x * y = z.$
2. $x, y$ and $z$ are different digits and $y > x$.

We need to find minimum x the least possible value of x, and select for Maximum x the maximum possible value of x.

For minimum x:
Options are 1, 2, 3, 4, 5 and 6.
So, it would be tempting to choose 1.
But remember, any number multiplying with 1 will give us same number as result.
So, it$ x = 1$, then
$1 * y = z $
$y = z $
This does not follow condition that y and z are different digits.

Let's try x = 2,
Then equation will be
$2y = z$
And we know y cannot be 1, as y > x.
So, y will always be a positive number and z will never be same as y.

So, x = 2 is min

For maximum x:
Starting with 6,
equation will be
$6y = z$
y cannot be 1 as y > z so y will never equal to z.

So, x = 6 is max.

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18 Jul 2024, 23:56

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$x*y=z$ & $y>x$

$\text{x, y, z}$ are different digits, hence $\text{x, y, z}<10$

when $x=1$, $y\geq{2}$, $z\geq{2}$ <10

when $x=2$, $y\geq{3}$, $z\geq{6}$ <10

when $x=3$, $y\geq{4}$, $z\geq{12}$ >10

Minimum x: 1
Maximum x: 2

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19 Jul 2024, 00:50

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Since XYZ are distinct digits, it will range from 0 to 9

If X is one then Y and Z will be equal, hence not one
If X is equals to 2 and Y is equals to 3, we get Z equals to 6 all distinct digits, hence minimum X is 2

Highest digit that can be considered is 9. but for nine, we have to multiply 3×3 and in that case XNY will be again equal, hence nine cannot be considered

Next highest digit is eight, which can be obtained by multiplying 4 x2 and we will get all distinct digits, hence maximum X is 4

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19 Jul 2024, 00:57

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A student was given an equation, x * y = z, where x, y, and z are different digits and y > x. Select for Minimum x the least possible value of x, and select for Maximum x the maximum possible value of x. Make only two selections, one in each column.

Minimum x = 2
Maximum x = 6

y>x ;x*y=z; x, y, z are different digits => x can't be 1 => Minimum x = 2; => Maximum x = 6

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19 Jul 2024, 01:01

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Bunuel

y > x
so the possible digits are

x y z
2 3 6
2 4 8

x cannot be 1 or 0 since x, z and y cannot be same
Minimum possible value of x is 2.

x cannot be greater than 2 since the least possible values will give us
z = 3*4 = 12 ....12 is not a digit
Maximum possible value of x is 2.

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19 Jul 2024, 01:09

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Minumum x is 2, because if it is 1, $y=z$
Maximum x is 6. For example, $x=6$, $y=7$, $z=42$

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19 Jul 2024, 01:12

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1) x*y=z
2) x,y,z are different digits
3) y>x

Minimum x can't be 1 since 1*y=y=z. THis is not possible due to condition (2)
Therefore, min x = 2

If x>=2 then x*y will no longer be a single digit #
eg. 3*4=12
Max value of x = 2

Left column B; Right Cloumn B

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19 Jul 2024, 02:46

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A student was given an equation, x * y = z, where x, y, and z are different digits and y > x. Select for Minimum x the least possible value of x, and select for Maximum x the maximum possible value of x. Make only two selections, one in each column.

As we know x,y, and z are different digits hence they can take values from 0 to 9.

Min x=
X*y=z (Can only be wtih 2X3=6 or 2*4=8). Hence 2.

Max x=
x*y=z (x Cannot be 3 as 3*4=12, Hence Max x=2).Hence 2.