Each of X₁, X₂ and X₃ represents a nonzero digit of the 3-digit base M positive integer X₁X₂X₃; where X₁, X₂ and X₃ are not necessarily distinct.

Determine the possible positive integer values of M, with 7 ≤ M ≤ 206, such that this equation has at least one valid solution.

X₁^X₂ + X₂^X₃ + X₃^X₁ = X₁X₂X₃ ± 6

Note: X₁X₂X₃ denotes the concatenation of the three digits.

M = {7, 8, 9, 11, 12, 14, 15, 17, 19, 20, 22, 23, 25, 28, 30, 62, 118, 126}

126₇   = 1² + 2⁶ + 6¹ -2
325₇   = 3² + 2⁵ + 5³
334₇   = 3³ + 3⁴ + 4³
343₇   = 3⁴ + 4³ + 3³ +6
513₇   = 5¹ + 1³ + 3⁵ +6
134₈   = 1³ + 3⁴ + 4¹ +6
253₈   = 2⁵ + 5³ + 3² +5
272₈   = 2⁷ + 7² + 2² +5
442₈   = 4⁴ + 4² + 2⁴ +2
732₈   = 7³ + 3² + 2⁷ -6
153₉   = 1⁵ + 5³ + 3¹
227₉   = 2² + 2⁷ + 7² +6
235₁₁  = 2³ + 3⁵ + 5² +4
244₁₁  = 2⁴ + 4⁴ + 4² +2
335₁₁  = 3³ + 3⁵ + 5³ +6
613₁₁  = 6¹ + 1³ + 3⁶ +4
163₁₂  = 1⁶ + 6³ + 3¹ -1
228₁₂  = 2² + 2⁸ + 8² -4
135₁₄  = 1³ + 3⁵ + 5¹ -6
144₁₄  = 1⁴ + 4⁴ + 4¹ -5
291₁₄  = 2⁹ + 9¹ + 1² -3
128₁₅  = 1² + 2⁸ + 8¹ -2
362₁₅  = 3⁶ + 6² + 2³ -6
254₁₇  = 2⁵ + 5⁴ + 4² -6
363₁₇  = 3⁶ + 6³ + 3³
183₁₉  = 1⁸ + 8³ + 3¹
329₂₀  = 3² + 2⁹ + 9³ -1
245₂₂  = 2⁴ + 4⁵ + 5² -4
193₂₃  = 1⁹ + 9³ + 3¹ +6
525₂₅  = 5² + 2⁵ + 5⁵ -2
373₂₈  = 3⁷ + 7³ + 3³ -6
145₃₀  = 1⁴ + 4⁵ + 5¹ -5
146₆₂  = 1⁴ + 4⁶ + 6¹ -5
755₁₁₈ = 7⁵ + 5⁵ + 5⁷ +6
147₁₂₆ = 1⁴ + 4⁷ + 7¹ -5

Posted by Dej Mar on 2011-04-21 18:42:08