Kth Smallest Amount With Single Denomination Combination
<p>You are given an integer array <code>coins</code> representing coins of different denominations and an integer <code>k</code>.</p>

<p>You have an infinite number of coins of each denomination. However, you are <strong>not allowed</strong> to combine coins of different denominations.</p>

<p>Return the <code>k<sup>th</sup></code> <strong>smallest</strong> amount that can be made using these coins.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block" style="
    border-color: var(--border-tertiary);
    border-left-width: 2px;
    color: var(--text-secondary);
    font-size: .875rem;
    margin-bottom: 1rem;
    margin-top: 1rem;
    overflow: visible;
    padding-left: 1rem;
">
<p><strong>Input:</strong> <span class="example-io" style="
    font-family: Menlo,sans-serif;
    font-size: 0.85rem;
">coins = [3,6,9], k = 3</span></p>

<p><strong>Output:</strong> <span class="example-io" style="
    font-family: Menlo,sans-serif;
    font-size: 0.85rem;
"> 9</span></p>

<p><strong>Explanation:</strong> The given coins can make the following amounts:<br />
Coin 3 produces multiples of 3: 3, 6, 9, 12, 15, etc.<br />
Coin 6 produces multiples of 6: 6, 12, 18, 24, etc.<br />
Coin 9 produces multiples of 9: 9, 18, 27, 36, etc.<br />
All of the coins combined produce: 3, 6, <u><strong>9</strong></u>, 12, 15, etc.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block" style="
    border-color: var(--border-tertiary);
    border-left-width: 2px;
    color: var(--text-secondary);
    font-size: .875rem;
    margin-bottom: 1rem;
    margin-top: 1rem;
    overflow: visible;
    padding-left: 1rem;
">
<p><strong>Input:</strong><span class="example-io" style="
    font-family: Menlo,sans-serif;
    font-size: 0.85rem;
"> coins = [5,2], k = 7</span></p>

<p><strong>Output:</strong><span class="example-io" style="
    font-family: Menlo,sans-serif;
    font-size: 0.85rem;
"> 12 </span></p>

<p><strong>Explanation:</strong> The given coins can make the following amounts:<br />
Coin 5 produces multiples of 5: 5, 10, 15, 20, etc.<br />
Coin 2 produces multiples of 2: 2, 4, 6, 8, 10, 12, etc.<br />
All of the coins combined produce: 2, 4, 5, 6, 8, 10, <u><strong>12</strong></u>, 14, 15, etc.</p>
</div>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= coins.length &lt;= 15</code></li>
	<li><code>1 &lt;= coins[i] &lt;= 25</code></li>
	<li><code>1 &lt;= k &lt;= 2 * 10<sup>9</sup></code></li>
	<li><code>coins</code> contains pairwise distinct integers.</li>
</ul>
Bit Manipulation; Array; Math; Binary Search; Combinatorics; Number Theory
class Solution:
    def findKthSmallest(self, coins: List[int], k: int) -> int:
        def check(mx: int) -> bool:
            cnt = 0
            for i in range(1, 1 << len(coins)):
                v = 1
                for j, x in enumerate(coins):
                    if i >> j & 1:
                        v = lcm(v, x)
                        if v > mx:
                            break
                m = i.bit_count()
                if m & 1:
                    cnt += mx // v
                else:
                    cnt -= mx // v
            return cnt >= k

        return bisect_left(range(10**11), True, key=check)
