Find the smallest number which when divided by 24 and 36 leaves a remainder of 11<\/p>\n\n\n\n
The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n To find the smallest number that leaves a remainder of 11 when divided by 24 and 36, we need to solve a system of congruences. Specifically, the number must satisfy the following conditions:<\/p>\n\n\n\n This means that the number xx when divided by both 24 and 36 should leave a remainder of 11 in both cases.<\/p>\n\n\n\n For the first congruence x\u226111 (mod 24)x \\equiv 11 \\ (\\text{mod} \\ 24), the number xx can be written as: x=24k+11x = 24k + 11<\/p>\n\n\n\n where kk is an integer.<\/p>\n\n\n\n For the second congruence x\u226111 (mod 36)x \\equiv 11 \\ (\\text{mod} \\ 36), substitute x=24k+11x = 24k + 11 into this equation: 24k+11\u226111 (mod 36)24k + 11 \\equiv 11 \\ (\\text{mod} \\ 36)<\/p>\n\n\n\n Simplifying this: 24k\u22610 (mod 36)24k \\equiv 0 \\ (\\text{mod} \\ 36)<\/p>\n\n\n\n This means that 24k24k must be divisible by 36.<\/p>\n\n\n\n Since 24k24k must be divisible by 36, divide both sides of the equation by 12: 2k\u22610 (mod 3)2k \\equiv 0 \\ (\\text{mod} \\ 3)<\/p>\n\n\n\n This means that kk must be divisible by 3. So, let k=3mk = 3m, where mm is an integer.<\/p>\n\n\n\n Substitute k=3mk = 3m into x=24k+11x = 24k + 11: x=24(3m)+11=72m+11x = 24(3m) + 11 = 72m + 11<\/p>\n\n\n\n Thus, the general solution for xx is: x=72m+11x = 72m + 11<\/p>\n\n\n\n To find the smallest number, set m=0m = 0 (since we want the smallest value): x=72(0)+11=11x = 72(0) + 11 = 11<\/p>\n\n\n\n Thus, the smallest number that satisfies both conditions is x=11x = 11.<\/p>\n\n\n\n The smallest number that when divided by 24 and 36 leaves a remainder of 11 is 11.<\/p>\n","protected":false},"excerpt":{"rendered":" Find the smallest number which when divided by 24 and 36 leaves a remainder of 11 The correct answer and explanation is: To find the smallest number that leaves a remainder of 11 when divided by 24 and 36, we need to solve a system of congruences. Specifically, the number must satisfy the following conditions: […]<\/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-46511","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/46511","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=46511"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/46511\/revisions"}],"predecessor-version":[{"id":46512,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/46511\/revisions\/46512"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=46511"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=46511"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=46511"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\n
Step 1: Express the number in terms of a general solution<\/h3>\n\n\n\n
Step 2: Solve for kk<\/h3>\n\n\n\n
Step 3: Substitute k=3mk = 3m back into the equation for xx<\/h3>\n\n\n\n
Step 4: Find the smallest solution<\/h3>\n\n\n\n
Conclusion<\/h3>\n\n\n\n