# Problem of the Day

Does positive integer have positive integer factors and such that and ?

(1) When is divided by , the remainder is

(2)

Reveal Answer### Solution

[latexpage]

If the factors are $a$ and $b$ such that $a-b=1$, they must be consecutive integers. By examining the products of all possible pairs of consecutive digits, we can conclude that the units digit of the product of any two consecutive integers can only be $0,2,$ or $6$.

Statement 1: This tells us that the unit’s digit of $n$ is 3. from the above analysis, we can say for sure that $n$ does not have a pair of consecutive integer factors. SUFFICIENT.

Statement 2: Again, the unit’s digit is 3, so 19,073 must not have a pair of consecutive integer factors. SUFFICIENT.

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