Justification 8468

The natural number n has at least 73 two-digit divisors. Prove that one of them is the number 60. Also, give an example of the number n, which has exactly 73 double-digit divisors, including a proper justification.

Correct answer:

n₁ = 11 * 12 * 13 * 14 * 16 * 17 * 18 * 19 * 21 * 22 * 23 * 24 * 26 * 27 * 28 * 29 * 31 * 32 * 33 * 34 * 36 * 37 * 38 * 39 * 41 * 42 * 43 * 44 * 46 * 47 * 48 * 49 * 51 * 52 * 53 * 54 * 56 * 57 * 58 * 59 * 61 * 62 * 63 * 64 * 66 * 67 * 68 * 69 * 71 * 72 * 73 * 74 * 76 * 77 * 78 * 79 * 81 * 82 * 83 * 84 * 86 * 87 * 88 * 89 * 91 * 92 * 93 * 94 * 96 * 97 * 98 * 99 * 15

Step-by-step explanation:

60 = 2^2 * 3 * 5

d₁ = 11 ....... 11 not div 5
d₂ = 12 ....... 12 not div 5 ; 12 div 3; 12 div 4
d₃ = 13 ....... 13 not div 5
d₄ = 14 ....... 14 not div 5
d₅ = 16 ....... 16 not div 5 ; 16 div 4
d₆ = 17 ....... 17 not div 5
d₇ = 18 ....... 18 not div 5 ; 18 div 3
d₈ = 19 ....... 19 not div 5
d₉ = 21 ....... 21 not div 5 ; 21 div 3
d₁₀ = 22 ....... 22 not div 5
d₁₁ = 23 ....... 23 not div 5
d₁₂ = 24 ....... 24 not div 5 ; 24 div 3; 24 div 4
d₁₃ = 26 ....... 26 not div 5
d₁₄ = 27 ....... 27 not div 5 ; 27 div 3
d₁₅ = 28 ....... 28 not div 5 ; 28 div 4
d₁₆ = 29 ....... 29 not div 5
d₁₇ = 31 ....... 31 not div 5
d₁₈ = 32 ....... 32 not div 5 ; 32 div 4
d₁₉ = 33 ....... 33 not div 5 ; 33 div 3
d₂₀ = 34 ....... 34 not div 5
d₂₁ = 36 ....... 36 not div 5 ; 36 div 3; 36 div 4
d₂₂ = 37 ....... 37 not div 5
d₂₃ = 38 ....... 38 not div 5
d₂₄ = 39 ....... 39 not div 5 ; 39 div 3
d₂₅ = 41 ....... 41 not div 5
d₂₆ = 42 ....... 42 not div 5 ; 42 div 3
d₂₇ = 43 ....... 43 not div 5
d₂₈ = 44 ....... 44 not div 5 ; 44 div 4
d₂₉ = 46 ....... 46 not div 5
d₃₀ = 47 ....... 47 not div 5
d₃₁ = 48 ....... 48 not div 5 ; 48 div 3; 48 div 4
d₃₂ = 49 ....... 49 not div 5
d₃₃ = 51 ....... 51 not div 5 ; 51 div 3
d₃₄ = 52 ....... 52 not div 5 ; 52 div 4
d₃₅ = 53 ....... 53 not div 5
d₃₆ = 54 ....... 54 not div 5 ; 54 div 3
d₃₇ = 56 ....... 56 not div 5 ; 56 div 4
d₃₈ = 57 ....... 57 not div 5 ; 57 div 3
d₃₉ = 58 ....... 58 not div 5
d₄₀ = 59 ....... 59 not div 5
d₄₁ = 61 ....... 61 not div 5
d₄₂ = 62 ....... 62 not div 5
d₄₃ = 63 ....... 63 not div 5 ; 63 div 3
d₄₄ = 64 ....... 64 not div 5 ; 64 div 4
d₄₅ = 66 ....... 66 not div 5 ; 66 div 3
d₄₆ = 67 ....... 67 not div 5
d₄₇ = 68 ....... 68 not div 5 ; 68 div 4
d₄₈ = 69 ....... 69 not div 5 ; 69 div 3
d₄₉ = 71 ....... 71 not div 5
d₅₀ = 72 ....... 72 not div 5 ; 72 div 3; 72 div 4
d₅₁ = 73 ....... 73 not div 5
d₅₂ = 74 ....... 74 not div 5
d₅₃ = 76 ....... 76 not div 5 ; 76 div 4
d₅₄ = 77 ....... 77 not div 5
d₅₅ = 78 ....... 78 not div 5 ; 78 div 3
d₅₆ = 79 ....... 79 not div 5
d₅₇ = 81 ....... 81 not div 5 ; 81 div 3
d₅₈ = 82 ....... 82 not div 5
d₅₉ = 83 ....... 83 not div 5
d₆₀ = 84 ....... 84 not div 5 ; 84 div 3; 84 div 4
d₆₁ = 86 ....... 86 not div 5
d₆₂ = 87 ....... 87 not div 5 ; 87 div 3
d₆₃ = 88 ....... 88 not div 5 ; 88 div 4
d₆₄ = 89 ....... 89 not div 5
d₆₅ = 91 ....... 91 not div 5
d₆₆ = 92 ....... 92 not div 5 ; 92 div 4
d₆₇ = 93 ....... 93 not div 5 ; 93 div 3
d₆₈ = 94 ....... 94 not div 5
d₆₉ = 96 ....... 96 not div 5 ; 96 div 3; 96 div 4
d₇₀ = 97 ....... 97 not div 5
d₇₁ = 98 ....... 98 not div 5
d₇₂ = 99 ....... 99 not div 5 ; 99 div 3