(1): Consider the set of the smallest 6 odd primes: {3,5,7,11,13,17}.
What is the largest multiple of this product which has
distinct digits ?
(2) What is the maximum number of digits a square-free integer (whether even or odd) can have if its digits are all distinct?
(3) What is the largest odd square-free integer with distinct digits having exactly n prime factors for n = 1,2,3,4,5? You can extend this to larger numbers of factors if you wish.
Note: a square-free integer is one whose prime factorization has exactly one factor for each prime that appears in it.
Part 1
The product is p= 4849845.
and 63047985 = 13 * p
found by
p=prod(primes(19))/2
t=p;
while t<9876543210
t=t+p;
good=true;
n=char(string(t));
for i=2:length(n)
f=strfind(n,n(i)) ;
if f(1)<i
good=false;
break
end
end
if good
disp([t t/p])
end
end
Part 2:
clearvars, clc
allDigs='9876543210'; didPrime=false;
largen=0; largeEven=0;
for digs=9:9
combs=nchoosek(allDigs,digs);
for i=1:length(combs)
p=perms(combs(i,:));
for j = 1:length(p)
n=str2num(p(j,:));
f=factor(n);
if length(f)==1
if didPrime==false
disp([n f])
didPrime=true;
end
else
if f(1)~=2
if n>largen
if length(unique(f))==length(f)
disp([f n])
largen=n;
end
end
else
if n>largeEven
if length(unique(f))==length(f)
disp([f n])
largeEven=n;
end
end
end
end
end
end
end
finds
2029 486769 987654301
2 5 98765431 987654310
meaning
product of primes 2029 and 486769 is 987654301
Including evens, the product 2 * 5 * 98765431 = 987654310
Part 3:
clearvars, clc
allDigs='9876543210'; didPrime=false;
largen=zeros(1,9); largeEven=zeros(1,9);
for digs=9:-1:8
combs=nchoosek(allDigs,digs);
for i=1:length(combs)
p=perms(combs(i,:));
for j = 1:length(p)
n=str2num(p(j,:));
f=factor(n);
lf=length(f);
if lf==1
if n>largen(1)
largen(1)=n;
end
else
if f(1)~=2
if lf<10
if n>largen(lf)
if length(unique(f))==length(f)
disp([f n])
largen(lf)=n;
end
end
end
else
if lf<10
if n>largeEven(lf)
if length(unique(f))==length(f)
disp([f n])
largeEven(lf)=n;
end
end
end
end
end
end
end
end
disp(largen)
disp(largeEven)
largest odd with n prime factors:
n
1 987654103 prime
2 987654301 2029 486769
3 987654201 3 23 14313829
4 987652403 41 53 61 7451
5 987652043 7 17 29 137 2089
6 987630215 5 11 13 17 193 421
7 987462105 3 5 7 23 37 43 257
8 820614795 3 5 7 11 13 31 41 43
9 none found with either 9 or 8 digits
The product of the first 9 odd primes is already a 10-digit number: 3234846615.
While evens were not asked for, here are even solutions also:
1 no 9-digit even numbers with only one prime factor (all have 2)
2 987653042 2 493826521
3 987654310 2 5 98765431
4 987654302 2 347 661 2153
5 987654230 2 5 73 419 3229
6 987650314 2 7 23 97 103 307
7 987534210 2 3 5 13 23 89 1237
8 987452610 2 3 5 19 29 31 41 47
9 857146290 2 3 5 7 11 13 17 23 73
It took 2209 seconds = 36 min 49 sec to run.
Edited on July 23, 2021, 2:18 pm
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Posted by Charlie
on 2021-07-23 14:16:24 |
computer solution
