Divisor Mods

Computer Science Level 3

Denote $S(n)$ the sum of all the positive divisors of $n$ , including itself. For example: $S(12)=1+2+3+4+6+12=28$ .

Find the smallest positive integer $n>10$ such that

$S(n) \equiv2018 \pmod n.$

The answer is 26.

1 solution

Zeeshan Ali
May 12, 2018

def S(n):
    '''
    INPUT:
        n: natual number
    OUTPUT:
        Sum: sum of all the complete-devisors of 'n'
    '''
    Sum = n
    for i in range(1, n//2+1):
        if n%i == 0:
            Sum += i
    return Sum

# Find the smallest natual number such that S(n) = 2018 (mod n)
for n in range(1, 9999):
    if S(n)%n == 2018:
        print ("S(", n ,") =", S(n), '=', 2018, "(mod", n, ')')
        break

S( 26) = 42 = 2018 (mod 26)

nice
here is a Mathematica code

n=1;While[Mod[DivisorSigma[1,n],n++]!=2018];n-1

Giorgos K. - 3 years ago

Isn't the statement true for $n=26$ ? The number of divisors is $1+2+13+26=42$ . But $42\equiv16 \pmod{26}$ and $2018\equiv16 \pmod{26}$ . Does the residue class need to be specified by the lowest positive element? My condition ignoring this fact is just the following:

1	`if (sumdiv(n) % n) == (2018 % n):`

Poca Poca - 3 years ago

I agree with you.

Ricardo Moritz Cavalcanti - 3 months, 2 weeks ago

Divisor Mods

The answer is 26.

1 solution

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