Problem 56050. Easy Sequences 73: Emergence of Fibonacci Insects

Created by Ramon Villamangca

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Cicadas are natural mathematicians. In order to avoid their natural predators (in this case, the wasp), they emerge from hibernation in \"a prime number\" of years. For example, if the emergence period of a wasp is every </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>12</w:t></w:r></w:customXml><w:r><w:t> years, selecting </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>4</w:t></w:r></w:customXml><w:r><w:t> or </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>6</w:t></w:r></w:customXml><w:r><w:t> or even </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>9</w:t></w:r></w:customXml><w:r><w:t> years emergence period for the cicadas, means that their population would be wiped out in a few years. But, by selecting a prime emergence period, say </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>11</w:t></w:r></w:customXml><w:r><w:t> years, they are assured not to meet their predators for a very long time. In this case, if cicadas and wasps emerged the same time today, it would take another </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>132</w:t></w:r></w:customXml><w:r><w:t> years for them to again see each other.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>In a theoretical environment, a particular species of cicadas selects to emerge every </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>F_n</w:t></w:r></w:customXml><w:r><w:t> (</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t>-th </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Fibonacci_number\"><w:r><w:rPr><w:b/></w:rPr><w:t>Fibonacci number</w:t></w:r></w:hyperlink><w:r><w:t>) days, while the predator wasps selects emergence period of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>F_m</w:t></w:r></w:customXml><w:r><w:t> (</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>m</w:t></w:r></w:customXml><w:r><w:t>-th Fibonacci number) days. </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>Given the values of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>m</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, and assuming that the insects emerge at the same time, today, in how many days will they again emerge together.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>For example, if the cicadas selected </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>F_{10} = 55</w:t></w:r></w:customXml><w:r><w:t>, and the wasps chose </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>F_8 = 21</w:t></w:r></w:customXml><w:r><w:t>, they will emerge at the same time again in </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>55\\times21=1155</w:t></w:r></w:customXml><w:r><w:t> days.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>NOTE: </w:t></w:r><w:r><w:t>You are given: </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>F_1=1</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>F_2=1</w:t></w:r></w:customXml><w:r><w:t>. Since the answer can be a huge number, please present your answer \"modulo </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>102334155</w:t></w:r></w:customXml><w:r><w:t>\".</w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 325px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 162.5px; transform-origin: 407px 162.5px; vertical-align: baseline; "><div style="block-size: 126px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 63px; text-align: left; transform-origin: 384px 63px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 377px 8px; transform-origin: 377px 8px; unicode-bidi: normal; ">Cicadas are natural mathematicians. In order to avoid their natural predators (in this case, the wasp), they emerge from hibernation in "a prime number" of years. For example, if the emergence period of a wasp is every <img src="data:image/png;base64,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" style="width: 18px; height: 18px;" width="18" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 53.5px 8px; transform-origin: 53.5px 8px; unicode-bidi: normal; "> years, selecting 4<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "> or 6<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 8px; transform-origin: 28px 8px; unicode-bidi: normal; "> or even 9<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 329px 8px; transform-origin: 329px 8px; unicode-bidi: normal; "> years emergence period for the cicadas, means that their population would be wiped out in a few years. But, by selecting a prime emergence period, say <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACQAAAAkCAYAAADhAJiYAAAA3ElEQVRYR2NkGGSAcZC5h2HUQYRiZDSEqB1CSkADPwLxW0IGY5EnSi+xUQYyrASIM4HYAIgvkuAgkvQScpAw0OJmqENgbiDWQWTpxecgL6ALdID4ChA3ArEJ1EXEOIhsvYRCCBYqZUBGJwkOQo5RkvSOOohQhhgNodEQwlJgjuYyQrXIaAiNhhAoBEhKB2hBRpLe0ZKaGiU1qMW3C4iVoVFRDqS7CKVkqDzJevFFGciwZCA2xmL5PahPZ+JwGNl6iU1DRAYI5cpGHUQoDEdDaDSECIUAIfnRNEQohAAxvlgl02j4IwAAAABJRU5ErkJggg==" style="width: 18px; height: 18px;" width="18" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 212.5px 8px; transform-origin: 212.5px 8px; unicode-bidi: normal; "> years, they are assured not to meet their predators for a very long time. In this case, if cicadas and wasps emerged the same time today, it would take another <img src="data:image/png;base64,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" style="width: 25.5px; height: 18px;" width="25.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 77px 8px; transform-origin: 77px 8px; unicode-bidi: normal; "> years for them to again see each other.</div><div style="block-size: 86px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 43px; text-align: left; transform-origin: 384px 43px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 266.5px 8px; transform-origin: 266.5px 8px; unicode-bidi: normal; ">In a theoretical environment, a particular species of cicadas selects to emerge every <img src="data:image/png;base64,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" style="width: 17px; height: 20px;" width="17" height="20"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4.5px 8px; transform-origin: 4.5px 8px; unicode-bidi: normal; "> (n<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.5px 8px; transform-origin: 10.5px 8px; unicode-bidi: normal; ">-th <a target='_blank' href = "https://en.wikipedia.org/wiki/Fibonacci_number">Fibonacci number</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 23.5px 8px; transform-origin: 23.5px 8px; unicode-bidi: normal; ">) days, while the predator wasps selects emergence period of <img src="data:image/png;base64,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" style="width: 19.5px; height: 20px;" width="19.5" height="20"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4.5px 8px; transform-origin: 4.5px 8px; unicode-bidi: normal; "> (m<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 90.5px 8px; transform-origin: 90.5px 8px; unicode-bidi: normal; ">-th Fibonacci number) days. <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 71px 8px; transform-origin: 71px 8px; unicode-bidi: normal; ">Given the values of n<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17px 8px; transform-origin: 17px 8px; unicode-bidi: normal; "> and m<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 367.5px 8px; transform-origin: 367.5px 8px; unicode-bidi: normal; ">, and assuming that the insects emerge at the same time, today, in how many days will they again emerge together.</div><div style="block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 115px 8px; transform-origin: 115px 8px; unicode-bidi: normal; ">For example, if the cicadas selected <img src="data:image/png;base64,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" style="width: 56px; height: 20px;" width="56" height="20"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 74.5px 8px; transform-origin: 74.5px 8px; unicode-bidi: normal; ">, and the wasps chose <img src="data:image/png;base64,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" style="width: 51px; height: 20px;" width="51" height="20"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 137.5px 8px; transform-origin: 137.5px 8px; unicode-bidi: normal; ">, they will emerge at the same time again in <img src="data:image/png;base64,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" style="width: 98px; height: 18px;" width="98" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19px 8px; transform-origin: 19px 8px; unicode-bidi: normal; "> days.</div><div style="block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 23px 8px; transform-origin: 23px 8px; unicode-bidi: normal; ">NOTE: <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 47.5px 8px; transform-origin: 47.5px 8px; unicode-bidi: normal; ">You are given: <img src="data:image/png;base64,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" style="width: 43.5px; height: 20px;" width="43.5" height="20"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "> and <img src="data:image/png;base64,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" style="width: 43.5px; height: 20px;" width="43.5" height="20"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 252px 8px; transform-origin: 252px 8px; unicode-bidi: normal; ">. Since the answer can be a huge number, please present your answer "modulo <img src="data:image/png;base64,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" style="width: 70.5px; height: 18px;" width="70.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4.5px 8px; transform-origin: 4.5px 8px; unicode-bidi: normal; ">".</div></div></div>

Cicadas are natural mathematicians. In order to avoid their natural predators (in this case, the wasp), they emerge from hibernation in "a prime number" of years. For example, if the emergence period of a wasp is every  years, selecting  or  or even  years emergence period for the cicadas, means that their population would be wiped out in a few years. But, by selecting a prime emergence period, say  years, they are assured not to meet their predators for a very long time. In this case, if cicadas and wasps emerged the same time today, it would take another  years for them to again see each other.
In a theoretical environment, a particular species of cicadas selects to emerge every  (-th Fibonacci number) days, while the predator wasps selects emergence period of  (-th Fibonacci number) days. Given the values of  and , and assuming that the insects emerge at the same time, today, in how many days will they again emerge together.
For example, if the cicadas selected , and  the wasps chose , they will emerge at the same time again in  days.
NOTE: You are given:  and . Since the answer can be a huge number, please present your answer "modulo ".

Solve

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Last Solution submitted on Feb 19, 2023

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Tim on 28 Sep 2022

I am getting 51840212 for test 7 instead of 90928381.

Ramon Villamangca on 29 Sep 2022

Hi Tim,

I've updated test 7. Please try again. Thanks.

Tim on 29 Sep 2022

Thanks, I would try, but I am using "BigInteger" and "for" and will therefore fail test 10.

Ramon Villamangca on 29 Sep 2022

Hi Tim, please consider that modulus of Fibonacci numbers are periodic...

GeeTwo on 8 Dec 2022

This should be part of the problem statement (test case 10):
Not Allowed in solution:
'persistent' 'global' 'BigInteger' 'java'
(OK so far, but also)
'if' 'for' 'switch' 'while' 'try'

Ramon Villamangca on 9 Dec 2022

Hi Gee, thanks for the comment. However, I don't think there is need to include it in the problem statement coz players can already clearly see the blocked functions in test 10.

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Problem 56050. Easy Sequences 73: Emergence of Fibonacci Insects

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Problem 56050. Easy Sequences 73: Emergence of Fibonacci Insects

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