Understand the problem

34!=295232799cd96041408476186096435ab000000 Find a,b,c,d (all single digits).

Source of the problem

BMO 2002

Topic
Number Theory
Difficulty Level
Easy
Suggested Book
An Excursion in Mathematics

Start with hints

Do you really need a hint? Try it first!

Get prepared to find the residue of 34! modulo various divisors! The substitution x=\frac{34!}{10^7}=<span><span class="mrow" id="MathJax-Span-300"><span class="mn" id="MathJax-Span-304">295232799</span><span class="mi" id="MathJax-Span-305">c</span><span class="mi" id="MathJax-Span-306">d</span><span class="mn" id="MathJax-Span-307">9604140809643</span><span class="mi" id="MathJax-Span-308">a</span><span class="mi" id="MathJax-Span-309">b should help simplify the calculation.
Note that x is divisible by 2^7. Find all possible residues of a number modulo 10^7 given that the number is divisible by 2^7. That’ll help you prove that a=5, b=2.
Note that x is divisible by 9. This gives that c+d is either 3 or 12.
As x is also divisible by 11, we must have c-d=-3 or 8. As 2c has to be an even integer less than or equal to 18, we must have c+d=3, c-d=-3. This gives c=0, d=3.

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