Question 1210666: How many different positive values of x will make this statement true: there are exactly 2 positive two-digit multiples of x.
Found 2 solutions by KMST, ikleyn:
Answer by KMST(5416)   (Show Source):
You can put this solution on YOUR website! I am going to assume, that "multiple" of x, means 2x, or 3x, or 4x, or any higher multiple but that x itself does not count as a multiple of x.
Then if double and triple the number (2x, and 3x) has two digits, there are at least 2 positive two-digit multiples of x.
However, if quadruple the number (4x) is also a positive two-digit number, then there would be at least 3 positive two-digit multiples of x, and that is not "exactly 2 positive two-digit multiples of x."
So, I want  but  .
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If we assume x must also be a whole number, the solution requires counting all the whole numbers from 25 (included) to 33.
There are  such numbers.
Answer by ikleyn(53979)  (Show Source):
You can put this solution on YOUR website! .
How many different positive values of x will make this statement true:
there are exactly 2 positive two-digit multiples of x.
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As this problem is formulated/worded in the post, it reminds me a lame horse with 3 (three) legs.
It is because the problem's formulation in the post is mathematically incorrect.
The correct formulation should say at the very beginning that the numbers x under consideration
are positive INTEGER numbers. Then the problem will sound harmonically (as any true Math problem should sound).
So, in my solution below I will assume that the numbers x are integer.
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In Math, the multiples of number x are numbers
1x (= x), 2x, 3x, . . . and so on,
i.e. the numbers of the form n*x, where n is any positive integer (by the definition).
Notice that any number x is a multiple to itself with the factor of multiplicity 1 (one).
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Thus, they want you find integer numbers x such that x itself is a two-digit,
2x is still a two-digit, but 3x is just a three-digit number.
From it, it is clear that the maximum possible number x under the condition is 49, giving doubled value 98,
and the minimum possible number x under the condition is 34, giving tripled value 102.
Thus, the solutions to the problem are integer numbers from 34 to 49 inclusive.
In all, there are 49 - 34 + 1 = 16 integer numbers satisfying the imposed condition. ANSWER
Solved.
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