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Question

How many real numbers satisfy the relation [x] = 32 {x}.


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Solution

We can write the given relation as {x} = 23 [x]
We know 0 {x} 1. [x] takes only integer values.

So the values [x] can assume are 0 and 1. For any other value of [x],

.{x} = 23 [x] goes out of its range([0, 1)).

When [x] = 0, {x} = 0

x = [x] + {x}

= 0 + 0 = 0

When [x] = 1, {x} = 23
x = [x] + {x}
= 1 + 23
= 53
So the two values of x we got are 0 and 53


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