Puzzle for February 28, 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 non-negative integer.
* "B mod A" equals the remainder of (B ÷ A).
** A! is A-factorial.
Scratchpad
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Hint #1
Subtract (A × F) from both sides of eq.3: B + (A × F) – (A × F) = A × E – (A × F) which becomes B = A × E – (A × F) which may be written as eq.3a) B = A × (E – F) In eq.3a, replace E – F with C ÷ A (from eq.2): B = A × (C ÷ A) which makes B = C
Hint #2
In eq.2, replace C with B: B ÷ A = E – F Since E and F are integers, then E – F = an integer, which means eq.2a) B ÷ A = an integer
Hint #3
Since B ÷ A is an integer (from eq.2a), then remainder of (B ÷ A) = 0 which means B mod A = 0 In eq.5, substitute 0 for B mod A: D = 0
Hint #4
Substitute 0 for D in eq.1: 0 + E = A + F which becomes eq.1a) E = A + F
Hint #5
Substitute A + F for E (from eq.1a) in eq.3a: B = A × (A + F – F) which makes B = A × (A) which may be written as eq.3b) B = A²
Hint #6
Subtract F from both sides of eq.1a: E – F = A + F – F which becomes E – F = A Substitute 0 for D, and A for E – F in eq.6: A! = A + 0 + A which makes eq.6a) A! = 2×A
Hint #7
Since A ≠ 0 (from eq.2), divide both sides of eq.6a by A: A! ÷ A = 2×A ÷ A which means eq.6b) (A – 1)! = 2 According to the definition of factorials: eq.6c) 2! = 2 × 1 = 2
Hint #8
Substitute 2! for 2 (from eq.6c) into eq.6b: (A – 1)! = 2! which means A – 1 = 2 Add 1 to both sides: A – 1 + 1 = 2 + 1 which makes A = 3
Hint #9
Substitute 3 for A in eq.3b: B = 3² which makes B = 9 and also makes C = B = 9
Hint #10
The logarithmic equation eq.4 may be re-written as the following equivalent exponential equation: A ^ F = B ("A ^ F" means "A raised to the power of F") In the above equation, substitute 3 for A, and 9 for B: 3 ^ F = 9 which means F = 2
Solution
Substitute 3 for A, and 2 for F in eq.1a: E = 3 + 2 which makes E = 5 and makes ABCDEF = 399052