a) Prove that a number is divisible by \(8\) if and only if the number formed by its laast three digits is divisible by \(8\).
b) Can you find an analogous rule for \(16\)? What about \(32\)?
Look at this formula found by Euler: \(n^2 +n +41\). It has a remarkable property: for every integer number from \(1\) to \(21\) it always produces prime numbers. For example, for \(n=3\) it is \(53\), a prime. For \(n=20\) it is \(461\), also a prime, and for \(n=21\) it is \(503\), prime as well. Could it be that this formula produces a prime number for any natural \(n\)?
Denote by \(n!\) (called \(n\)-factorial) the following product \(n!=1\cdot 2\cdot 3\cdot 4\cdot...\cdot n\). Show that if \(n!+1\) is divisible by \(n+1\), then \(n+1\) must be prime. (It is also true that if \(n+1\) is prime, then \(n!+1\) is divisible by \(n+1\), but you don’t need to show that!)
A little Jimmy visited his four aunties today. Each of them prepared a cake for him and his parents. Auntie Martha made a carrot cake, Auntie Camilla made a sponge, Auntie Becky made a chocolate cake and Auntie Anne made a fudge. Jimmy would like to visit the aunties the next time when aunties all make the same cakes again. What day should he pick, if Auntie Martha makes a carrot cake every two days, Auntie Camilla makes a sponge every three days, Auntie Becky makes a chocolate cake every four days and Auntie Anne makes a fudge every seven days?
a) Two numbers, \(a\) and \(b\), are relatively prime and their product is equal to \(3^5 \times 7^2\). What could these numbers be? Find all the possibilities.
b) The gcd of two numbers, \(c\) and \(d\), is \(20\) and their product is \(2^4 \times 5^3\). What could these numbers be? Find all the answers.
Two numbers are \(a = 2 \times 3^5 \times 31^2 \times 7\) and \(b= 7^2 \times 2^4 \times 3^2 \times 29^2\). Find their greatest common divisor and least common multiple.
A valiant adventurer enters a dragon’s cave looking for the Holy Grail. She knows that Holy Grail is a chalice that is tall, made of gold, has encrusted rubies, and has an ancient inscription written on it. Upon entering, the knight discovers a long corridor of chalices, all marked with natural numbers starting from 1. He examines the first chalices and discovers, that every 10th chalice is tall, every 15th is made of gold, every 28th has encrusted rubies and every 27th has an ancient description. Assuming that is universally true for all the chalices in the cave, which chalice should the adventurer check so she doesn’t waste too much time checking all of them?
The gcd of numbers \(a\) and \(b\) is \(40\). What can be their smallest possible product? What could be their greatest possible product?
a) While visiting Cape Verde, Pirate Jim and Pirate Bob bought several chocolate chip cookies each. Jim paid 93 copper coins for his cookies and Bob paid 102 copper coins. What could be the price of a single cookie if it is a natural number?
b) Captain Hook and Captain Kid bought several tricorn hats each. Captain Hook paid 6 silver coins more than Captain Kid. What could be the price of a tricorn hat if it is an integer?
Show that \(a-b\) is divisible by the greatest common divisor of \(a\) and \(b\).