If A is an integer, what is the value of A?

If A is an integer, what is the value of A = ?

S1:
1, 000, 000 < (23)^A < 10,000,000

(10)^6 < (23)^A < (10)^7

A positive integer (23) raised to an exponent must be between two positive integers as per statement 1. Thus, exponent A must be a positive integer.

assume (in an alternate universe) (23)^k is equal to the next lowest power of the lower boundary:

Assume (23)^k = (10)^5 = 100,000

If we were to increase the power by +1, the next consecutive integer, we would end up with:

(23)^(k + 1) = 2, 300, 000 ——- a number that would fall within the boundaries of the inequality

However, if we were to increase the power by +1 more to the next consecutive integer, we would end up with a value TOO HIGH for the given range. One can see this is true by multiplying by *(10), a value less than *(23)

2,300,000 * (10) = 23, 000, 000 < (23)^(k + 2)

and

1, 000, 000 < (23)^A < 10, 000, 000 < 23, 000, 000

What this tells us is that it would be impossible to have two consecutive powers of (23) fall within the range:
(10)^6 < (23)^A < (10)^7

In fact, if we start from the assumption that there IS two consecutive powers of 23 that fall within the range and assume:

(23)^k = just a tiny bit larger than 1, 000, 000 (which is the lower boundary of the range)

and then we were to add just one more (+1) power of 23 to the product, we would end up with:

(1, 000, 000) * (23) = 23,000,000 ————a number FAR outside the upper boundary of 10, 000, 000 and invalid.

Therefore, it would be impossible for two powers of (23) —in which A is an integer — to lie between the range of 1 million and 10 million.

Thus, there can be only one positive integer exponent of (23) that falls between this range. We do not need to necessarily know this value of the exponent A, just that one unique value of A exists.

Statement 1 is sufficient

S2: the information given by statement 2 is a tautology: it will be true for any value of A. It is similar to writing 1 = 1

Dive both sides of the equation by (7)^A

(7^2 — 1) / 48 = 1

49 — 1 = 48

48 = 48

Any integer value of A is possible

S2 not sufficient

*A*

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