Puzzle for November 21, 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.
* BC is a 2-digit number (not B×C).
** "A mod B" equals the remainder of (A ÷ B).
Scratchpad
Help Area
Hint #1
Add B to both sides of eq.3: B + D + B = C – B + B which becomes eq.3a) 2×B + D = C In eq.2, replace C with 2×B + D (from eq.3a): 2×B + D + F = D + E Subtract D from each side of the equation above: 2×B + D + F – D = D + E – D which becomes eq.2a) 2×B + F = E
Hint #2
In eq.1, replace E with 2×B + F (from eq.2a): A + B + F = 2×B + F Subtract B and F from both sides of the equation above: A + B + F – B – F = 2×B + F – B – F which simplifies to A = B
Hint #3
In eq.6, substitute B for A: F = B mod B which means F = remainder (B ÷ B) which means F = 0
Hint #4
Substitute B for A in eq.5: E = B × B which makes E = B²
Hint #5
Substitute 0 for F, and B² for E in eq.2a: 2×B + 0 = B² which makes 2×B = B² Since B ≠ 0 (from eq.6), divide both sides of the equation above by B: 2×B ÷ B = B² ÷ B which makes 2 = B and makes A = B = 2 and also makes E = B² = 2² = 4
Hint #6
Substitute 2 for B in eq.3: 2 + D = C – 2 Add 2 to both sides of the above equation: 2 + D + 2 = C – 2 + 2 which makes eq.3a) D + 4 = C
Hint #7
eq.4 may be written as: D ^ A = 10×B + C – E Substitute 2 for A and B, D + 4 for C (from eq.3a) and 4 for E in the above equation: D ^ 2 = 10×2 + D + 4 – 4 which becomes D² = 20 + D Subtract both 20 and D from each side of the equation above: D² – 20 – D = 20 + D – 20 – D which becomes D² – 20 – D = 0 which may be written as eq.4a) D² – D – 20 = 0
Hint #8
eq.4a is a quadratic equation in standard form. Using the quadratic equation solution formula to solve for D in eq.4a yields: D = { (–1)×(–1) ± sq.rt.[(–1)² – (4 × (1) × (–20))] } ÷ (2 × (1)) which becomes D = {1 ± sq.rt.[1 – (–80)]} ÷ 2 which becomes D = {1 ± sq.rt.(81)} ÷ 2 which becomes D = (1 ± 9) ÷ 2 In the above equation, either D = (1 + 9) ÷ 2 = 10 ÷ 2 = 5 or D = (1 – 9) ÷ 2 = –8 ÷ 2 = –4 Since D must be a non-negative integer, then D ≠ –4 and therefore makes D = 5
Solution
Substitute 5 for D in eq.3a: 5 + 4 = C which makes 9 = C and makes ABCDEF = 229540