1.
Let 1

*,*4*, . . .*and 9*,*16*, . . .*be two arithmetic progressions. The set*S*is the union of the first 2004 terms of each sequence. How many distinct numbers are in*S*?
2.
Given a sequence of six strictly increasing positive integers such that each number
(besides the first) is a multiple of the one before it and the sum of all six
numbers is 79, what is the largest number in the sequence?

3.
What is the largest positive integer

*n*for which*n*^{3}+ 100 is divisible by*n*+ 10?
4.
A positive integer is written on each face of a cube. Each vertex is then assigned
the product of the numbers written on the three faces intersecting the vertex.
The sum of the numbers assigned to all the vertices is equal to 1001. Find the
sum of the numbers written on the faces of the cube.

5.
Call a number

*prime looking*if it is composite but not divisible by 2, 3, or 5. The three smallest prime-looking numbers are 49, 77, and 91. There are 168 prime numbers less than 1000. How many prime-looking numbers are there less than 1000?
6.
A positive integer

*k*greater than 1 is given. Prove that there exist a prime*p*and a strictly increasing sequence of positive integers*a*1*,**a*2*, . . . ,**an**, . . .*such that the terms of the sequence*p*+

*ka*1

*,*

*p*+

*ka*2

*, . . . ,*

*p*+

*kan*

*, . . .*

are
all primes.

7.
Given a positive integer

*n*, let*p**(**n**)*be the product of the nonzero digits of*n*. (If*n*has only one digit, then*p**(**n**)*is equal to that digit.) Let*S*=

*p*

*(*1

*)*+

*p*

*(*2

*)*+· · ·+

*p*

*(*999

*).*

What
is the largest prime factor of

*S*?
8.
Let

*m*and*n*be positive integers such that
lcm

*(**m**,**n**)*+ gcd*(**m**,**n**)*=*m*+*n**.*
Prove
that one of the two numbers is divisible by the other.

9.
Let

*n*= 2^{31}3^{19}. How many positive integer divisors of*n*^{2}are less than*n*but do not divide*n*?
10.
Show that for any positive integers

*a*and*b*, the number*(*36

*a*+

*b*

*)(*

*a*+ 36

*b*

*)*

cannot
be a power of 2.

11.
Compute the sum of the greatest odd divisor of each of the numbers 2006, 2007,

*. . .*, 4012.
12.
Compute the sum of all numbers of the form

*a**/**b*, where*a*and*b*are relatively prime positive divisors of 27000.
Sumber
: Buku

**104 Number Theory Problems**
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