Writing natural numbers in base b. For any natural number b> 1 we say nEN is...

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Writing natural numbers in base b. For any natural number b> 1 we say nEN is written base b if there are co, C1, C2,... such that 0≤ c < b and n = o cib'. We call the c; the digits of n in base b. (a) Write the digits of 144 in base 2, 5, 10 and 12. (b) Prove that any n N can be written in base b. (c) Prove the following statement. If n is written in base b as n = Σio Cibi = i-o dibi, where 0 < ck, de <b, then k = l and c; = dį. (d) Prove that for any n E N only finitely many digits of n in base b are nonzero. (e) Recall that n! II. The factorial digits of a natural number n are the numbers d₁, d2, .. ... EN such that n = i dii! where 0≤ di ≤i. Find the factorial digits for the numbers 1, 2, 3, 4, 5 (f) * Prove that the factorial digits are uniquely defined for any n E N. 1 Writing natural numbers in base b. For any natural number b> 1 we say nEN is written base b if there are co, C1, C2,... such that 0≤ c < b and n = o cib'. We call the c; the digits of n in base b. (a) Write the digits of 144 in base 2, 5, 10 and 12. (b) Prove that any n N can be written in base b. (c) Prove the following statement. If n is written in base b as n = Σio Cibi = i-o dibi, where 0 < ck, de <b, then k = l and c; = dį. (d) Prove that for any n E N only finitely many digits of n in base b are nonzero. (e) Recall that n! II. The factorial digits of a natural number n are the numbers d₁, d2, .. ... EN such that n = i dii! where 0≤ di ≤i. Find the factorial digits for the numbers 1, 2, 3, 4, 5 (f) * Prove that the factorial digits are uniquely defined for any n E N. 1

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