Three positive integers (a, b, c) with a<b<c are called a
Pythagorean triple if the sum of the square of a and the square of
b is equal to the square of c. Write a program that prints all
Pythagorean triples (one in a line) with a, b, and c all smaller
than 1000, as well the total number of such triples in the end.
Arrays are not allowed to appear in your code. Hint: user nested
loops (Can you...
8.Let a and b be integers and d a positive
integer.
(a) Prove that if d divides a and d divides b, then d divides both
a + b and a − b.
(b) Is the converse of the above true? If so, prove it. If not,
give a specific example of a, b, d showing
that the converse is false.
9. Let a, b, c, m, n be integers. Prove that if a divides each of b
and c,...
Suppose that two stacks of positive integers are needed for a
project. Each stack is to contain integers that are less than or
equal to 500. One stack is to contain even integers; the other
stack is to contain odd integers. Also, the total number of values
in the combined stacks at any given time will not exceed 200. ▪
Design a way to implement both stacks in only one(pay attention!!!)
1-D array. ▪ Write a Python class definition(s) for...
Question#1
How many positive integers between 100 and 888 inclusive,
a) are divisible by 7?
b) are odd?
c) have distinct digits?
d) are not divisible by 6?
e) are divisible by either 4 or 7?
f) are not divisible by either 4 or 7?
g) are divisible by 4 but not by 7?
h) are divisible by 4 and 7?
Question#1
How many positive integers between 100 and 888 inclusive,
a) are divisible by 7?
b) are odd?
c)...
Let a and b be positive integers, and let d be their greatest
common divisor. Prove that there are infinitely many integers x and
y such that ax+by = d. Next, given one particular solution x0 and
y0 of this equation, show how to find all the solutions.
The least common multiple of nonzero integers a and b is the
smallest positive integer m such that a|m and b|m. It is denoted
[a, b], or sometimes [a, b] for short. Prove the following:
1) If a|k and b|k, then [a, b]|k.
2) If gcd(a, b) = 1, then [a, b] =ab
3) If c >0,then [ca, cb] =c·[a, b].
4) If a >0 and b >0,then [a, b] =ab / gcd(a, b).