Puzzle for October 11, 2020 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
* "C ^ E" means "C raised to the power of E". "E ^ C" means "E raised to the power of C".
** BC is a 2-digit number (not B×C).
*** "F mod B" equals the remainder of (F ÷ B).
Scratchpad
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Hint #1
In eq.1, replace E with B ÷ C (from eq.4): A × B ÷ C = B × C Multiply both sides of the above equation by C: A × B ÷ C × C = B × C × C which becomes A × B = B × C² Since B ≠ 0 (from eq.6), divide both sides of the equation above by B: A × B ÷ B = B × C² ÷ B which makes A = C²
Hint #2
In eq.2, substitute C² for A: eq.2a) C ^ E = C² To make eq.2a true, either: C = 0, or C = 1, or E = 2 Since C ≠ 0 (from eq.4), then either: C = 1, or E = 2
Hint #3
Begin checking: C = 1 ... Substituting 1 for C in eq.4 would yield: B ÷ 1 = E which would make B = E
Hint #4
Finish checking: C = 1 ... Substituting B for E in eq.6 would yield: F mod B = B which would mean remainder of (F ÷ B) = B However, remainder of (F ÷ B) must be < B which means F mod B ≠ B and therefore means C ≠ 1 making E = 2
Hint #5
Substitute 2 for E in eq.4: B ÷ C = 2 Multiply both sides of the equation above by C: B ÷ C × C = 2 × C which makes B = 2×C
Hint #6
Substitute 2 for E in eq.3: eq.3a) 2 ^ C = F Since C ≠ 0 or 1, then to make eq.3a true, check several possible values for C (≥ 2) and F: If C = 2, then 2 ^ 2 = F which makes F = 4 If C = 3, then 2 ^ 3 = F which makes F = 8 If C = 4, then 2 ^ 4 = F which makes F = 16 If C > 4, then F > 16 Since C and F must be one-digit integers, then C and F must be either: C = 2 and F = 4 or C = 3 and F = 8
Hint #7
eq.5 may be written as: 10×B + C = A × D Substitute (2×C) for B, and C² for A in the equation above: 10×(2×C) + C = C² × D which is equivalent to 20×C + C = C² × D which becomes 21×C = C² × D Divide both sides by C: 21×C ÷ C = C² × D ÷ C which becomes eq.5a) 21 = C × D
Solution
The only two non-negative integers for C and D that make eq.5a true are: C = 3 and D = 7 or C = 7 and D = 3 Since C must equal 2 or 3, then C = 3 and D = 7 making F = E ^ C = 2 ^ 3 = 8 (from eq.3) B = 2×C = 2 × 3 = 6 A = C² = 3² = 9 and ABCDEF = 963728