Modulus. 34 % 5 = 4, 3 % 4 = .75?

Careful — modulus (`%`) only works with **integers** in C++.
It gives the **remainder** after division, **not a fraction**.

👉 Examples:
– `34 % 5 = 4` ✅ (since 34 ÷ 5 = 6 remainder 4)
– `3 % 4 = 3` ✅ (since 3 ÷ 4 = 0 remainder 3)

If you want `.75`, that’s **floating-point division**:
“`cpp
3.0 / 4.0 = 0.75
“`

So:
– `%` → remainder (integers only).
– `/` → division (integers or floats, depends on type).

Want me to show you how modulus and division pair together to break money into coins (like your pennies example)?
How is 3%4 3?
Because `%` is remainder, not “percentage.”

Think of it this way:

– **3 ÷ 4** = 0 with remainder **3**
(since 4 doesn’t fit into 3 even once)

So:
“`cpp
3 % 4 == 3
“`

If you wanted **0.75**, you need **division with floats**:
“`cpp
3 / 4 // = 0 (integer division)
3.0 / 4 // = 0.75 (floating-point division)
“`

⚡ Rule of thumb:
– `%` → leftover after division (only works on integers).
– `/` → gives the quotient (integer or decimal depending on types).

Want me to show a quick table of examples (like `7 % 3`, `10 % 6`, etc.) so you see the pattern?
oh, it is asking after all of the ints are taken away – what remains
Exactly 👍

That’s the perfect way to think of `%`:

– First do **integer division** (how many whole times the divisor fits).
– Then `%` gives you **what’s left over**.

Examples:

– `10 % 3` → 10 ÷ 3 = 3 remainder **1**
– `25 % 7` → 25 ÷ 7 = 3 remainder **4**
– `3 % 4` → 4 doesn’t fit into 3 at all, so remainder is just **3**

So `%` is literally “what’s left after you peel off the whole numbers.”

Want me to also show how this connects back to your **coin change** problem?
so any number that is lower than the mod number is assigned that number as left over
💯 You nailed it.

If the **dividend** (the number on the left) is smaller than the **divisor** (the number on the right in `%`), then the divisor “doesn’t fit even once.”

That means:
“`cpp
x % y == x // whenever x < y
“`

Examples:
– `2 % 5 = 2`
– `7 % 10 = 7`
– `99 % 100 = 99`

Because the divisor can’t take away a whole chunk, the remainder is just the number itself.

👉 Do you want me to show how this is *useful* in practice (like wrapping around array indices or time on a clock)?
Got it
Perfect ✅ You’ve got the core idea of modulus down.

It’s one of those deceptively simple operators, but it pops up everywhere—coins, clocks, looping through arrays, even hash functions.

Want me to throw you a couple of “real-world” modulus puzzles to practice on, or do you feel solid enough for now?
define iomanip
“ is a C++ **header file** that gives you input/output **manipulators** — tools to format how numbers and text look when printed with `cout` or read with `cin`.

Think of it as “I/O formatting utilities.”

Common manipulators from “:

– `std::setw(n)` → set field **width** (pads with spaces if needed)
– `std::setprecision(n)` → set number of **decimal places**
– `std::fixed` → forces fixed-point notation (not scientific)
– `std::showpoint` → always show the decimal point (e.g., `5.000`)
– `std::setfill(ch)` → change the padding character

Example:
“`cpp
#include
#include
using namespace std;

int main() {
double pi = 3.14159;

cout << setw(10) << setfill('*') << pi << endl;
cout << fixed << setprecision(2) << pi << endl;
}
“`

Output:
“`
***3.14159
3.14
“`

⚡ So: “ = “pretty print” tools for your console I/O.

Want me to show you how this applies directly to your paycheck/travel program (so your numbers line up neatly)?