Elementary arithmetic, with brief notices of its history... by Robert Potts.

11
2. Express the composite numbers between 100 and 200 by the
products of their prime factors.
3. Write down the prime numbers contained between 1 and 100
and between 900 and 1000.
4. Shew whether prime numbers occur in any regular order in
each successive tens of the series of the natural numbers.
5. Decompose 831600 into its prime factors.
6. Find the factors of 6552 and of 4080, and their highest comlmon
factor.
7. Find the greatest common measure of 1071, 1092, 2310, by resolving the numbers into their prime factors.
8. Resolve 132288, 107328, 138216, and 97344, into their prime
factors. And find their greatest common measure and their least
common multiple.
9. The product of four consecutive numbers is 1680, find them.
10. When a series of numbers have been resolved into their prime
factors, which of these factors must be taken to form by their product
(1) the greatest common measure, (2) the least common multiple of
the numbers. Form the greatest common measure and the least
common multiple of 405, 570, 910.
11. Prove by any number of seven figures that when a number is
divided by 9, it will have the same remainder as the sum of the digits
divided by 9.
12. The product of the first and second of three numbers is 377, of
the second and third 481, and of the first and third 1073; find the
three numbers.
VII.
1. Every prime number greater than 5, increased or diminished by
unity, is divisible by 6. Exemplify the truth of this by any ten prime
numbers, and show whether the converse is true.
2. Find the prime factors of 1728, and shew in how many ways
1728 can be divided into two factors.
3. Determine whether either of the numbers 785432 and 785431
is divisible by 9.
4. Prove that one of the two numbers 23456789 and 23457698 is
divisible by 11, and the other is not.
5. Shew how to find the number of divisors of a given composite
number. What number of divisors has the least common multiple of
1428, 1287, 1560?
6. If the product of the natural numbers which precede any prime
number be increased by unity, the sum is exactly divisible by that
prime number. Ex. Shew that 1 x 2 x x 4 x 5 x 6 x 7 x 8 x 9 x 10+ 1
is exactly divisible by the prime number 11.
7. Any odd number not ending with 5 being given as a multiplicand; shew that it is always possible to find ahother number which