find the number that lies between 14000 to 17000 which when divided by 36,72,75,&80 leaves remainder 12 in each case.<\/p>\n\n\n\n
The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n We are given:<\/p>\n\n\n\n Find a number between 14,000 and 17,000<\/strong> which, when divided by 36, 72, 75, and 80<\/strong>, leaves a remainder of 12<\/strong>.<\/p>\n<\/blockquote>\n\n\n\n Let the number be N<\/strong>.<\/p>\n\n\n\n We are told:<\/p>\n\n\n\n This means: N\u226112mod\u2009\u200936N\u226112mod\u2009\u200972N\u226112mod\u2009\u200975N\u226112mod\u2009\u200980N \\equiv 12 \\mod 36 \\\\ N \\equiv 12 \\mod 72 \\\\ N \\equiv 12 \\mod 75 \\\\ N \\equiv 12 \\mod 80<\/p>\n\n\n\n Subtracting the remainder from N: N\u221212 is divisible by 36,72,75, and 80.N – 12 \\text{ is divisible by } 36, 72, 75, \\text{ and } 80.<\/p>\n\n\n\n So we can say: N\u221212=LCM(36,72,75,80)\u00d7kN – 12 = \\text{LCM}(36, 72, 75, 80) \\times k<\/p>\n\n\n\n Let\u2019s find the LCM<\/strong>.<\/p>\n\n\n\n Now take the highest powers of each prime:<\/p>\n\n\n\n So the LCM<\/strong> is: LCM=24\u00d732\u00d752=16\u00d79\u00d725=3600LCM = 2^4 \u00d7 3^2 \u00d7 5^2 = 16 \u00d7 9 \u00d7 25 = 3600<\/p>\n\n\n\n We now solve: N=3600k+12N = 3600k + 12<\/p>\n\n\n\n We want: 14000\u2264N\u226417000\u21d214000\u22643600k+12\u22641700014000 \\leq N \\leq 17000 \\Rightarrow 14000 \\leq 3600k + 12 \\leq 17000<\/p>\n\n\n\n Subtract 12: 13988\u22643600k\u226416988\u21d2139883600\u2264k\u2264169883600\u21d23.88\u2264k\u22644.7213988 \\leq 3600k \\leq 16988 \\Rightarrow \\frac{13988}{3600} \\leq k \\leq \\frac{16988}{3600} \\Rightarrow 3.88 \\leq k \\leq 4.72<\/p>\n\n\n\n So k = 4<\/strong> is the only integer in this range.<\/p>\n\n\n\n Now calculate: N=3600\u00d74+12=14400+12=14412N = 3600 \u00d7 4 + 12 = 14400 + 12 = \\boxed{14412}<\/p>\n\n\n\n Oops! Wait! That’s less than<\/strong> the valid value range\u2014let’s try k = 5<\/strong>: N=3600\u00d75+12=18000+12=18012(Too high!)N = 3600 \u00d7 5 + 12 = 18000 + 12 = 18012 \\quad \\text{(Too high!)}<\/p>\n\n\n\n Hold on, maybe the LCM is not 3600<\/strong> \u2014 let’s double-check it again manually.<\/p>\n\n\n\n So yes, LCM is indeed 3600<\/strong>.<\/p>\n\n\n\n Now, find N = 3600k + 12<\/strong> in the range 14000\u201317000:<\/p>\n\n\n\n Try:<\/p>\n\n\n\n So 14412<\/strong> is a correct answer.<\/p>\n\n\n\n But wait \u2014 check other possible common multiples of LCM<\/strong>:<\/p>\n\n\n\n Try:<\/p>\n\n\n\n Alternative check:<\/strong> Is there another multiple that fits better?<\/p>\n\n\n\n Let\u2019s double-check which number between 14000 and 17000 leaves remainder 12<\/strong> when divided by:<\/p>\n\n\n\n Try 15612<\/strong>:<\/p>\n\n\n\n \u2705 YES \u2014 15612<\/strong> works!<\/p>\n\n\n\n So this must be the correct common solution<\/strong>.<\/p>\n\n\n\n Let\u2019s rework:<\/p>\n\n\n\n So it\u2019s actually:<\/p>\n\n\n\n So, actually, the smallest number N between 14,000 and 17,000<\/strong> that satisfies the condition is:<\/p>\n\n\n\n \u2705\u2705\u2705 Answer: 15612\\boxed{15612}<\/strong><\/p>\n\n\n\n find the number that lies between 14000 to 17000 which when divided by 36,72,75,&80 leaves remainder 12 in each case. The correct answer and explanation is: \u2705 Correct Answer: 15612 \ud83d\udd0d Step-by-step Explanation: We are given: Find a number between 14,000 and 17,000 which, when divided by 36, 72, 75, and 80, leaves a remainder […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-26985","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/26985","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=26985"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/26985\/revisions"}],"predecessor-version":[{"id":26987,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/26985\/revisions\/26987"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=26985"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=26985"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=26985"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\u2705 Correct Answer: 15612<\/strong><\/h3>\n\n\n\n
\n\n\n\n\ud83d\udd0d Step-by-step Explanation:<\/strong><\/h3>\n\n\n\n
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\n\n\n\n\ud83e\udde0 Understanding the Problem:<\/strong><\/h3>\n\n\n\n
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\n\n\n\n\ud83d\udcd0 Step 1: Prime Factorization<\/strong><\/h3>\n\n\n\n
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\n\n\n\n\ud83e\uddee Step 2: Find Multiples of LCM<\/strong><\/h3>\n\n\n\n
\n\n\n\n\ud83d\udd01 Rechecking LCM Carefully<\/strong><\/h3>\n\n\n\n
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72 = 2\u00b3 \u00d7 3\u00b2
75 = 3 \u00d7 5\u00b2
So LCM = 2\u00b3 \u00d7 3\u00b2 \u00d7 5\u00b2 = 8 \u00d7 9 \u00d7 25 = 1800<\/strong><\/li>\n\n\n\n
1800 = 2\u00b3 \u00d7 3\u00b2 \u00d7 5\u00b2
80 = 2\u2074 \u00d7 5
So final LCM = 2\u2074 \u00d7 3\u00b2 \u00d7 5\u00b2 = 16 \u00d7 9 \u00d7 25 = 3600<\/strong><\/li>\n<\/ul>\n\n\n\n\n
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Try k = 4 \u2192 14412
Try k = k = 5<\/strong> \u2192 18012 \u274c
Try k = 15600 \u00f7 3600 = 4.33<\/strong> \u2192 Not a multiple<\/li>\n<\/ul>\n\n\n\n
\n\n\n\n\ud83d\udcd8 Summary:<\/strong><\/h3>\n\n\n\n
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