Let us define the positive number n with four
digits a,b,c and d such that n=abcd
with a,b,c,d∈Z, 1≤a≤9, 0≤b≤9,
0≤c≤9 and 0≤d≤9. Let us then say
that a cool number is a four digit number,
say n, such that the two digit numbers written as
ab and cd are given by ab=r×s and
cd=(r−1)×(s+1) for some non−negative integers
r and s, r≠s. For example, 8081 has
a=8,b=0 and 80=10×8= while
c=8,d=1 and 81=9×9=(10−1)(8+1).
So, for n=8081, r=10 while s=8.
How many n, for 1000≤n≤9999, are cool?
For n∈[1000,9999],n∈Z, how many n exist
so that ab=r×s and cd=r×s+1? Call
such n warm numbers.
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