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# IntDiv

```solidity
pragma circom 2.1.8;
include "../node_modules/circomlib/circuits/comparators.circom";

// Create a circuit that is satisfied if `numerator`,
// `denominator`, `quotient`, and `remainder` represent
// a valid integer division. You will need a comparison check, so
// we've already imported the library and set n to be 252 bits.
//
// Hint: integer division in Circom is `\`.
// `/` is modular division
// `%` is integer modulus

template IntDiv(n) {
    signal input numerator;
    signal input denominator;
    signal input quotient;
    signal input remainder;

}

component main = IntDiv(252);

```

### Answer

```solidity
template IntDiv(n) {
    signal input numerator;
    signal input denominator;
    signal input quotient;
    signal input remainder;

    // (quotient * divisor) + remainder = dividend
    // num/den = q + r -> num = (q * den) + r

    // constraint forwards
    numerator === (quotient * denominator) + remainder;
    
    
    signal a;
    a === LessThan(n)([remainder, denominator]);
    a === 1;
}

component main = IntDiv(252);
```

**The forward constraint on its own is insufficient. Why?**

`a === (q * b) + r`

Free to submit convenient values of r and a to appease equality.

**Example:**

`24 === (3 * 4) + 12` \
`24 === 12 + 12`

Therefore, must constraint r to be less than the dividend/numerator. \
This will result in the shrinking of solution set.