Find three different natural numbers, larger than \(100\) such that each of them is divisible by the difference of the other two numbers? The values of differences also have to be different from each other.
We know that the product \(c \times d\) is divisible by a prime \(p\). Show that either \(c\) or \(d\) must be divisible by \(p\).
Find the last two digits of the number \[33333333333333333347^4 - 11111111111111111147^4\]
Replace all stars with ”+” or ”\(\times\)” signs so the equation holds: \[1*2*3*4*5*6=100\] Extra brackets may be added if necessary. Please write down the expression into the answer box.
In how many ways can one change \(\pounds 2\) into coins worth \(50\)p, \(20\)p and \(10\)p? One does not necessarily need to use all available coin types, i.e. having \(5\) coins of \(20\)p and \(10\) coins of \(10\)p is allowed.
Convert the binary number \(10011\) into decimal, and convert the decimal number \(28\) into binary. Multiply by \(2\) as binary numbers both \(10011\) and the result of conversion of \(28\) into binary numbers.
The ternary numeral system has only \(3\) digits: \(0,\) \(1\) and \(2\). Therefore the number \(3\) is written in ternary as \(10\). Write down the numbers \(23\) and \(156\) in ternary and add them as ternary.
Write down the first fifteen numbers in binary system.
Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written as combinations of letters from the Latin alphabet, each letter with a fixed integer value:
I&V&X&L&C&D&M
1&5&10&50&100&500&1000
For example the first \(12\) numbers in Roman Numerals are written as: \(I,\,II,\, III,\, IV,\, V,\, VI,\, VII,\, VIII,\, IX,\, X,\, XI,\, XII\), where the notations \(IV\) and \(IX\) can be read as "one less than five" and "one less than ten" correspondingly. A number containing two or more decimal digits is built by appending the Roman numeral equivalent for each digit, from highest to lowest, as in the following examples: the current year \(2024\) as \(MMXXIV\), number \(17\) as \(XVII\) and number \(42\) as \(XLII\) or \(XXXXII\). Let’s see how to multiply Roman numerals by multiplying \(17\) and \(42\).
Write down in Roman numerals the numbers \(14\) and \(61\) and multiply them as Roman numerals.