Department of Math and Computer Science
Problems of the Month 1995

Proposed by Dan Jurca

What is the last non-zero digit in the decimal representation of 1,000,000! ?

Remark: Using Stirling's famous approximation one can show that

1,000,000!  »  8.26399×105,565,708,

and, using a standard result in number theory, one can show easily that the decimal representation of 1,000,000! ends with 249,998 zeros.

Therefore 1,000,000! is a number of 5,565,709 decimal digits, and if written out has the following appearance:

8,263,¼{ 5,315,706  more digits}¼,x00,000,¼{ 249,990 more zeros}¼,000

where x is a non-zero digit.

What is x?

The proposer used a computer to solve this problem, and you are permitted to do so also; the proposer hopes that someone finds a better solution.

Solution by the proposer

Notation: N={1,2,3,¼}; W={0}ÈN; P={2,3,5,7,11,13,¼}, the set of primes.

For each prime p let n_p:N® W as follows-

n Î NÞ n=2n2(n)×3n3(n)×5n5(n)×¼ = Õ p Î P  pnp(n);

i.e., n_p(n) is the number of occurrences of the prime p  in the factorization of n as a product of primes.

Clearly the decimal representation of n! ends with n₅(n!) zeros, and

n Î WÞ  n! 10n5(n!) =2n2(n!)-n5(n!)×3n3(n!)× Õ p Î P, 5 < p  pnp(n!).

Next one computes at once that if p º p₀ (mod 10) and 0 < k, then p^k º the value shown in the following table (mod 10). (k is taken modulo 4.)

p0   k º 0 k º 1 k º 2 k º 3  0   0 0 0 0  1   1 1 1 1  2   6 2 4 8  3   1 3 9 7  4   6 4 6 4  5   5 5 5 5  6   6 6 6 6  7   1 7 9 3  8   6 8 4 2  9   1 9 1 9

This suggests the following method, which is easily implemented as a computer program, for computing the right-most non-zero digit in the decimal representation of n!-

0.

Put x=2^{n₂(n!)-n₅(n!)}×3^n₃(n!) mod 10; one uses the information in the table above to do this easily.

1.

For each prime p such that 5 < p £ n one determines n_p(n!) and its congruence class mod 10. Again referring to the table above one multiplies x by the appropriate value, and of course reduces this to its value modulo 10.

2.

After accumulating this product (mod 10) for all primes £ n the value of x is the right-most non-zero digit in the decimal representation of n!.

Also solved by Professor Bill Nico and Devang Vora. One incorrect solution was received.