What is the value of 11^x-11^(x+2), where x is the largest integer

We have that \(30,030=2*3*5*7*11*13\) is divisible by \(11^x\) (where \(x\) is an integer). Now, ask yourself what can be the largest integer value of \(x\). Could it be 2 or more?

We have 11 ^ x - 11 ^x+2

Simplifying : 11 ^ x - 11^x * 11^2

Taking 11^x as common we get : 11^x (1 - 11^2)

--> 11^x * -120

The question asks us to find the max. value for x so that 11^x is a factor of 30030

Prime Factors of 30030 are 5 x 3 x 2 x 11 x 7 x 13..

Because 11 appears only once , therefore the maximum value that x can have is 1 ....

Using the value of x as 1 and adding it the equation we simplified we get

11 x -120 = - 1320 (B)

Here 30030/11 gives 2730. This cannot be divided further and hence 11 can have the power of only 1. substituting in the given equation gives the ans -1320.

Given
x is the largest integer 30030

Find the Value of
11^x - 11^(x + 2)

Simplify 11^x - 11^(x + 2) we get
(11^x)(1-11^2) -> (11^x)(1-11)(1+11) -> (11^x)(-10)(12)

Elimination (11^x)(-10)(12)
1.From this we know that the value has to be -ve
2.The units digit must be a 0.

Hence eliminate options A,C,E. Now we are left with B and D

Look at the equation again (11^x)(-10)(12) -> (11^x)(-120)
Now in option D we have -120 this can possible only if x is the above equation is 0. However we are told that 11^X is a factor of 30030. So x has to be greater than 0. Which eliminates the option D.

Answer is B

how did we infer that x=1?

11^x would be a factor of 30,030 even if x=0, since 11^0=1 and 1 is a factor of every integer. The point is that we are looking for the largest possible value of integer x such that 11^x is a factor of 30,030, which is for x=1.

Hope it's clear.

Agree. So just to make sure we can see if 30030 is divisible by 11. If it is then we really dont care by how much because we know that x is not 0 now. So the only option we are left with is now B.

30,030 has 11 as a factor
So largest value of x possible = 1

Just place x = 1 in the equation

11 - 1331 = -1320 = Answer = B

In 30030 there's only one 11, so x = 1.

11^x - 11^(x+2) = 11^x(1-11^2) = -120. 11^x

With x = 1 --> -120.11^1 = -1320 --> B

What is the value of 11^x-11^(x+2), where x is the largest integer such that 11^x is a factor of 30,030?

A. -1331
B. -1320
C. -121
D. -120
E. -1

Looks to be simple enough

11^x(1 - 11^2) = 11^x(1 - 121) = -120*11^x

now 11^x is a factor of 30,030 -------> is it divisible by 11
if not divisible by 11 then largest 11^0 = 1 will always be a factor of 30,030

Anywaz 30,030 is divisble by 11 (30030/11 = 2730)

Now the product of -120*11^x will be -ve and will end with a 0. Only answer choices B and D fit

D) If 11 was not divisible by 30030 then this could be it but since it is divisible by 11, it must be something higher (in this a lower value since its -ve)

hence ans is B)

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