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Question

Let n be the smallest composite number such that it can be written as the product of two positive integers that differ by 10. How many distinct prime factors does n have?


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Solution

Compute the number of prime factors.

Consider the positive integer as x and y.

According to the given condition

y-x=10y=x+10

Since n is the product of the positive integers ,use the first positive integers values for x to find the composite number n.

At x=0 ,we have

n=0(10+0)n=0.

0 is not a composite number.

At x=1,

we have n=1(1+10)=11..... which is again not a composite number.

At x=2,

we have n=2(2+10)=24......which is a composite number.

Therefore. n=24is the smallest composite number

Now, factors of 24 are 24=2×2×2×3.

Hence, the distinct prime factors are 2 and 3.


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