HOW OLD AM I?
Since becoming very old, I have sent annual birthday messages
to my colleagues declaring just how old. Here are some of them,
not in chronological order. Needless to say, my current age
is the largest of the integers described.
(A)
I'm the sum of two squares (both non-zero) in two different ways.
This is a new experience, so claim your flapjack in the coffee room.
(Of course, n^2 + n^2 counts as a sum of two squares.)
(B)
Now that I'm n, I'm prime for the pth time.
When I was p, I was prime for the qth time.
When I was q, I was prime for the rth time.
When I was r, I had exactly three divisors (including 1 and r
itself) - for the very first time.
When I was 1, I wasn't prime.
Tell me n at coffee time, and you may be awarded a sticky
crispy thing.
(C)
When I am divided by the sum of my digits, the remainder is 1.
It will be 35 years before this happens again, so claim
your flapjack in the coffee room today.
(D)
Certain numbers are products of two distinct primes, e.g 21 = 3x7.
n is the kth one.
k is the rth one.
r is a square.
What is n?
(You can assume that it's more than 25 and less than 100.)
(E)
Still no grandchildren (as far as I know) but now that I'm n,
I have more divisors than ever before, in fact, k of them.
k itself has k/2 divisors. It's not the only number that has
this property, but it's the bigger of the two that do.
(Of course you won't have forgotten that if p,q,r are prime, and
n = p^a q^b r^c, then n has (a+1)(b+1)(c+1) divisors,
including 1 and n itself.)
(F)
I'm back in my prime after six years.
It will be another six years before it happens again.
I'm not 150 yet.
(G)
I am the sum of two squares, also (for the second year
in succession) a product of two distinct primes.
This has happened once before, but it will be 88 years before it
happens again, so claim your sticky crispy thing at coffee time
today.
(H)
Define N(r,s) to be the rth number that is the product of s
distinct primes.
n = N(4,2) + N(2,3) = ?
As a check, n = N(r,2) for a certain r.
Correct answers rewarded by a sticky crispy thing at coffee time.