I'd suggest using a dedicated library for this, such as decimal.js.

Decimal objects have a useful method isInteger(), that can be invoked after the division is complete.

You can set the precision required, though I believe the default should suffice (20, see precision)

function isQuotientInteger(dividend, divisor) {
    return new Decimal(dividend).dividedBy(new Decimal(divisor)).isInteger();
}

let testInputs = [ { dividend: 1.15, divisor: 0.01 }, { dividend: 9.96, divisor: 0.01 }, { dividend: 15.121, divisor: 0.01 } , { dividend: 3.14159, divisor: 0.00001 }];

formatRow('Dividend', 'Divisor', 'Quotient is integer')
for(let testInput of testInputs) {
    formatRow(testInput.dividend, testInput.divisor, isQuotientInteger(testInput.dividend, testInput.divisor))
}

function formatRow(...row) {
    console.log(...row.map(f => (f + '').padEnd(10)))
}
   

.as-console-wrapper { max-height: 100% !important; top: 0; }

<script src="https://cdnjs.cloudflare.com/ajax/libs/decimal.js/10.3.1/decimal.min.js" integrity="sha512-Ou4M+sSU8oa+mE3juYqR3JmW633MUpMhe1cd+IusOtfjkMo8I3zXs4fRmjmCFqpRg5RK/geqoXBY8XRwFY2Rsg==" crossorigin="anonymous" referrerpolicy="no-referrer"></script>

Borrowing from this thread:

Rounding to the nearest hundredth of a decimal in JavaScript

let ex1=1.15 / 0.01
let ex2=9.96 / 0.01
let ex3=15.121 / 0.01
const desired_place=3

function round(num, places) {
    //round # to however many places you need
    let multiplier = Math.pow(10, places);
    let rounded= Math.round(num * multiplier) / multiplier;
    return Number.isInteger(rounded)? true : false
}

console.log(round(ex1,desired_place))  //true
console.log(round(ex2,desired_place))  //true
console.log(round(ex3,desired_place))  /false

In Javascript all numbers are internally represented as floating point numbers, which can in many cases lead to visible or invisible rounding issues, when mathematical operations are executed. It is unfortunate that Number.isInteger() produces an unexpected result in the cases you list, where we humans can clearly "see" that the result of the operation must be an integer value. Here is a practical solution to find out if a quotient is an integer with a given precision:

First you should use Number.toFixed() to get a number rounded to your required precision, and then apply Number.toInteger().

function isInteger(arg) {
  const precision = 12; // for example
  return Number.isInteger(Number(arg.toFixed(precision)));
}

console.log(isInteger(1.15 / 0.01));
console.log(isInteger(9.96 / 0.01));
console.log(isInteger(15.121 / 0.01));