A list of remainders can look like a set of unrelated conditions. Sometimes they are all saying the same thing in disguise.
The Remainder Code
Riddle statement
I am looking for the smallest positive integer with this property:
- when divided by 2, it leaves remainder 1;
- when divided by 3, it leaves remainder 2;
- when divided by 4, it leaves remainder 3;
- when divided by 5, it leaves remainder 4;
- when divided by 6, it leaves remainder 5;
- when divided by 7, it leaves remainder 6.
What is it?
Show solution
Solution
Answer: the smallest number is 419.
The number leaves remainder 1 when divided by 2, remainder 2 when divided by 3, remainder 3 when divided by 4, and so on.
That means it is always exactly 1 short of being divisible by each of those numbers.
If we call the desired number $N$, then:
$
N+1
$
must be divisible by 2, 3, 4, 5, 6, and 7.
The smallest positive number divisible by all of them is their least common multiple:
$
\operatorname{lcm}(2,3,4,5,6,7)=420
$
Therefore:
$
N+1=420
$
and so:
$
N=419.
$