2.3 KiB
id | title | challengeType | forumTopicId | dashedName |
---|---|---|---|---|
5900f3b31000cf542c50fec6 | Problem 71: Ordered fractions | 5 | 302184 | problem-71-ordered-fractions |
--description--
Consider the fraction, \frac{n}{d}
, where n
and d
are positive integers. If n
< d
and highest common factor, {{HCF}(n, d)} = 1
, it is called a reduced proper fraction.
If we list the set of reduced proper fractions for d
≤ 8 in ascending order of size, we get:
\frac{1}{8}, \frac{1}{7}, \frac{1}{6}, \frac{1}{5}, \frac{1}{4}, \frac{2}{7}, \frac{1}{3}, \frac{3}{8}, \frac{\textbf2}{\textbf5}, \frac{3}{7}, \frac{1}{2}, \frac{4}{7}, \frac{3}{5}, \frac{5}{8}, \frac{2}{3}, \frac{5}{7}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \frac{7}{8}
It can be seen that \frac{2}{5}
is the fraction immediately to the left of \frac{3}{7}
.
By listing the set of reduced proper fractions for d
≤ limit
in ascending order of size, find the numerator of the fraction immediately to the left of \frac{3}{7}
.
--hints--
orderedFractions(8)
should return a number.
assert(typeof orderedFractions(8) === 'number');
orderedFractions(8)
should return 2
.
assert.strictEqual(orderedFractions(8), 2);
orderedFractions(10)
should return 2
.
assert.strictEqual(orderedFractions(10), 2);
orderedFractions(9994)
should return 4283
.
assert.strictEqual(orderedFractions(9994), 4283);
orderedFractions(500000)
should return 214283
.
assert.strictEqual(orderedFractions(500000), 214283);
orderedFractions(1000000)
should return 428570
.
assert.strictEqual(orderedFractions(1000000), 428570);
--seed--
--seed-contents--
function orderedFractions(limit) {
return true;
}
orderedFractions(8);
--solutions--
function orderedFractions(limit) {
const fractions = [];
const fractionValues = {};
const highBoundary = 3 / 7;
let lowBoundary = 2 / 7;
for (let denominator = limit; denominator > 2; denominator--) {
let numerator = Math.floor((3 * denominator - 1) / 7);
let value = numerator / denominator;
if (value > highBoundary || value < lowBoundary) {
continue;
}
fractionValues[value] = [numerator, denominator];
fractions.push(value);
lowBoundary = value;
}
fractions.sort();
return fractionValues[fractions[fractions.length - 1]][0];
}