Practice problems 1 - pending

Q1. Consider the equation ab + bc + ca = abc - a - b - c. Is it necessary for all of a,b,c to be even?

Solution:
Yes.
Consider the cases one by one when it is not so.

Case 1:
All odd.
Rewrite the equation like this:
a(b+1) + b(c+1) + c(a+1) = abc
RHS = odd
LHS = even + even + even = even
Not possible.

Case 2:
a,b odd. c even. WLOG

Rewrite the equation like this:
a(b+1) + b(c+1) + c(a+1) = abc

RHS = even
LHS = even + odd + even = odd
Not possible.

Case 3:
a odd. b,c even. WLOG

Rewrite the equation like this:
a(b+1) + b(c+1) + c(a+1) = abc

RHS = even
LHS = odd + even + even = odd
Not possible.

So we have checked all cases: 1 odd, 2 odd, 3 odd. None of them works.
Only option is all even.

Q2. Sum of all divisors of a natural number?

Q3. If last 2 digits of 107n and n are same then what is the smallest value of n?
Solution:
107n = n mod 100
=> 106n = 0 mod 100
=> 6n = 0 mod 100
=> 6n = 100k
=> 3n = 50k
=> n has to be a multiple of 50 since 3 is not.
So smallest n = 50.

Q4. Find the product of all positive divisors of 44.

Q5. Find the sum of all divisors (including negative divisors) of 1463.




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