# When a number is divided by 2, 3, 4, 5 or 6, we always get a remainder of 1. But on dividing a number by 7, we find the number is divisible? What is the smallest such number? What are other such numbers? and how to solve such problems?

##### 1 Answer

301 is the lowest, with other numbers being in the form of 301 + a multiple of 420 and a general solution below:

#### Explanation:

I read this problem and haven't been able to let it go - thanks for asking it (although I may not sleep tonight...)

We're looking for a number that is divisible by 2, 3, 4, 5, and 6 (meaning that N is a multiple of those numbers) and that number +1 being divisible by 7.

We can make progress on this by seeing that the Lowest Common Multiple of these numbers is $2 \times 2 \times 3 \times 5 = 60$. Now the key is to find a multiple of 60 such that when we add 1 to it, the number becomes divisible by 7.

After some trial and error, I found that the lowest number that works is 300, which is $5 \times 60$. When we take $300 + 1 = 301 , 301 \div 7 = 43$.

With some more trial and error, I found that the next multiples of 60 increase by 7, so 5, 12, 19,... and that the resulting multiple of 7 increases by a factor of 60, so 43, 103, 163, etc.

To summarize so far, we have:

$\left(\begin{matrix}\text{factor with 60" & "multiple" & "+1" & "/7" \\ 5 & 300 & 301 & 43 \\ 12 & 720 & 721 & 103 \\ 19 & 1140 & 1141 & 163 \\ vdots & vdots & vdots & vdots \\ "+7" & "+420" & "+420" & "+60} \\ \vdots & \vdots & \vdots & \vdots\end{matrix}\right)$

So we have the smallest number (301) and other such numbers (721. 1141, etc). And we have a pattern we can follow for other questions of this sort:

• find the LCM of the numbers where there is a remainder

• find the lowest occurrence where, by adding the remainder to the LCM, it's evenly divisible by the other number (my method being trial and error)

• find the pattern of factors (which appears to be additive, using the factor of the "other side" - the "factor along with 60" increasing by 7, while the "/7" increases by 60).