Puzzle for April 10, 2021 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit positive integer.
* "C ^ B" means "C raised to the power of B".
** "E mod B" equals the remainder of (E ÷ B).
Scratchpad
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Hint #1
In eq.3, replace F with A + B + C (from eq.1): B + D = A + B + C – C which becomes B + D = A + B Subtract B from both sides of the above equation: B + D – B = A + B – B which makes D = A
Hint #2
In eq.2, replace D with A: A + E = C + A Subtract A from each side of the equation above: A + E – A = C + A – A which makes E = C
Hint #3
In eq.4, substitute E for C: D – E + F = E + E which becomes D – E + F = 2×E Add E to each side of the above equation: D – E + F + E = 2×E + E which becomes eq.4a) D + F = 3×E
Hint #4
eq.5 may be written as: C ^ B = D + F + E Substitute E for C, and 3×E for D + F (from eq.4a) in the equation above: E ^ B = 3×E + E which becomes E ^ B = 4×E Divide both sides by E: E ^ B ÷ E = 4×E ÷ E which becomes eq.5a) E ^ (B–1) = 4
Hint #5
In eq.5a, E must be an even number (an odd number raised to the power of a non-negative integer cannot equal 4). To make eq.5a true, then: E = 2 and B = 3 making 2 ^ (3–1) = 2 ^ 2 = 4 or E = 4 and B = 2 making 4 ^ (2–1) = 4 ^ 1 = 4
Hint #6
Check: E = 2, and B = 3 ... Substituting 2 for E, 3 for B, and A for D in eq.6 would yield: 2 mod 3 = A - A which would make 2 mod 3 = 0 However, 2 mod 3 = 2 which means 2 mod 3 ≠ 0 and means E ≠ 2 and B ≠ 3 therefore making E = 4 and B = 2 and also makes C = E = 4
Hint #7
Substitute 4 for E in eq.4a: D + F = 3×4 which makes D + F = 12 Subtract D from both sides of the above equation: D + F – D = 12 – D which makes eq.4b) F = 12 – D
Hint #8
Substitute 2 for B, 12 – D for F (from eq.4b), and 4 for C in eq.3: 2 + D = 12 – D – 4 which becomes 2 + D = 8 – D In the equation above, subtract 2 from each side, and add D to each side: 2 + D – 2 + D = 8 – D – 2 + D which becomes 2×D = 6 Divide both sides by 2: 2×D ÷ 2 = 6 ÷ 2 which makes D = 3 and also makes A = D = 3
Solution
Substitute 3 for D in eq.4b: F = 12 – 3 which makes F = 9 and ABCDEF = 324349