A positive integer n 1 is called beautiful if n can be written in one and only one way as n = a1 +

Question

A positive integer n > 1 is called beautiful if n can be written in one and only one way as n = a1 + a2 + … + ak = a1  a2 … ak for some positive integers a1, a2, …., ak, where k > 1 and a1  a2  ….  ak. (For example 6 is beautiful since 6 = 3  2  1 = 3 + 2 + 1; and this is unique. But 8 is not beautiful since 8 = 4 + 2 + 1 + 1 = 4  2  1  1 as well as 8 = 2 + 2 + 2 + 1 + 1 = 2  2  2  1  1, so uniqueness is lost.) Find the largest beautiful number less than 100.

Accepted Answer

{"content":[{"type":"paragraph","content":[{"marks":[{"type":"bold"}],"text":"Step 1:","type":"text"},{"text":" 99 = 9 × 11 × 1 × 1 × 1 × 1 ……………. × 1 (79 times ‘1’)","type":"text"}]},{"content":[{"text":"Step 2:","type":"text","marks":[{"type":"bold"}]},{"text":" = 9 + 11 + 1 + 1 + ----------- 79 times","type":"text"}],"type":"paragraph"},{"type":"paragraph","content":[{"marks":[{"type":"bold"}],"text":"Step 3:","type":"text"},{"text":" = 33 × 3 × 1 × 1 × 1 ….. (66 times 1)","type":"text"}]},{"content":[{"marks":[{"type":"bold"}],"text":"Step 4:","type":"text"},{"text":" = 33 + 3 + 1 + 1 + ……… 66 times","type":"text"}],"type":"paragraph"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 5:","type":"text"},{"text":" = Hence 99 is not beautiful.","type":"text"}],"type":"paragraph"},{"type":"paragraph","content":[{"marks":[{"type":"bold"}],"text":"Step 6:","type":"text"},{"text":" 98 = 49 × 2 × 1 × 1 × ……. 47 times","type":"text"}]},{"content":[{"marks":[{"type":"bold"}],"text":"Step 7:","type":"text"},{"type":"text","text":" = 49 + 2 + 1 + 1 + …….. 47 times"}],"type":"paragraph"},{"type":"paragraph","content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step 8:"},{"type":"text","text":" = 14 × 7 × 1 × 1 × …… 77 times"}]},{"type":"paragraph","content":[{"marks":[{"type":"bold"}],"text":"Step 9:","type":"text"},{"text":" = 14 + 7 + 1 + 1 + ……. 77 times","type":"text"}]},{"content":[{"marks":[{"type":"bold"}],"text":"Step 10:","type":"text"},{"text":" Hence 98 is not beautiful.","type":"text"}],"type":"paragraph"},{"content":[{"text":"Step 11:","type":"text","marks":[{"type":"bold"}]},{"text":" 97 is not beautiful as it can not be written in these expanded forms.","type":"text"}],"type":"paragraph"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 12:","type":"text"},{"text":" 96 = 2 × 48 × 1 × 1 × ….. 46 times","type":"text"}],"type":"paragraph"},{"type":"paragraph","content":[{"text":"Step 13:","type":"text","marks":[{"type":"bold"}]},{"type":"text","text":" = 2 + 48 + 1 + 1 + ……. 46 times"}]},{"content":[{"marks":[{"type":"bold"}],"text":"Step 14:","type":"text"},{"text":" = 3 × 32 × 1 × 1 × ……. 61 times","type":"text"}],"type":"paragraph"},{"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step 15:"},{"text":" = 3 + 32 + 1 + 1 + …… 61 times","type":"text"}],"type":"paragraph"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 16:","type":"text"},{"text":" Hence 96 is not beautiful.","type":"text"}],"type":"paragraph"},{"type":"paragraph","content":[{"marks":[{"type":"bold"}],"text":"Step 17:","type":"text"},{"type":"text","text":" 95 = 19 × 5 × 1 × 1 × ….. 71 times"}]},{"content":[{"text":"Step 18:","type":"text","marks":[{"type":"bold"}]},{"text":" = 19 + 5 + 1 + 1 + ……. 71 times","type":"text"}],"type":"paragraph"},{"content":[{"text":"Step 19:","type":"text","marks":[{"type":"bold"}]},{"text":" As 95 is uniquely represents hence it is beautiful.","type":"text"}],"type":"paragraph"}],"type":"doc"}

Guide / Hint

Solution