1978 IMO Problems/Problem 1

Problem

$m$ and $n$ are positive integers with $m < n$. The last three decimal digits of $1978^m$ are the same as the last three decimal digits of $1978^n$. Find $m$ and $n$ such that $m + n$ has the least possible value.

Solution

We have $1978^m\equiv 1978^n\pmod {1000}$, or $978^m-978^n=1000k$ for some positive integer $k$ (if it is not positive just do $978^n-978^m=-1000k$). Hence $978^n\mid 1000k$. So dividing through by $978^n$ we get $978^{m-n}-1=\frac{1000k}{978^n}$. Observe that $2\nmid LHS$, so $2\nmid RHS$. So since $2|| 978^n$, clearly the minimum possible value of $n$ is $3$ (and then $489^n\mid k$). We will show later that if $n$ is minimal then $m$ is minimal. We have $978^{m-3}-1\equiv 0\pmod {125}\Leftrightarrow 103^{m-3}\equiv 1\pmod {125}$. Hence, $m-3\mid \varphi(125)\Rightarrow m-3\mid 100$. Checking by hand we find that only $m-3=100$ works (this also shows that minimality of $m$ depends on $n$, as claimed above). So $m=103$. Consequently, $m+n=106$ with $\boxed{(m,n)=(103,3)}$.

The above solution was posted and copyrighted by cobbler and Andreas. The original thread for this problem can be found here: [1] and [2]

Video Solution

https://www.youtube.com/watch?v=SRl4Wnd60os

See Also

1978 IMO (Problems) • Resources
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First question
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Problem 2
All IMO Problems and Solutions
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