Puzzle for January 13, 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.
* CD is a 2-digit number (not C×D).
** "B mod F" equals the remainder of (B ÷ F).
Scratchpad
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Hint #1
Add D to both sides of eq.1: C – D + D = D + E + D which becomes eq.1a) C = 2×D + E In eq.2, replace C with 2×D + E (from eq.1a): A – D = 2×D + E – E which becomes A – D = 2×D Add D to both sides of the above equation: A – D + D = 2×D + D which makes eq.1b) A = 3×D
Hint #2
In eq.3, replace A with 3×D: D × F = 3×D Divide both sides of the equation above by D: D × F ÷ D = 3×D ÷ D which means F = 3
Hint #3
In eq.4, substitute 3 for F: E = D ÷ 3 Multiply both sides of the above equation by 3: E × 3 = D ÷ 3 × 3 which makes 3×E = D
Hint #4
Substitute (3×E) for D in eq.1b: A = 3×(3×E) which makes A = 9×E
Hint #5
Substitute (3×E) for D in eq.1a: C = 2×(3×E) + E which becomes C = 6×E + E which makes C = 7×E
Hint #6
eq.5 may be written as: A × B = 10×C + D – E Substitute 9×E for A, (7×E) for C, and 3×E for D into the equation above: 9×E × B = 10×(7×E) + 3×E – E which becomes 9×E × B = 70×E + 2×E which becomes 9×E × B = 72×E Divide both sides by 9×E: 9×E × B ÷ 9×E = 72×E ÷ 9×E which makes B = 8
Solution
In eq.6, substitute 8 for B, 3 for F, and 3×E for D: 8 mod 3 = 3×E – E which means remainder of (8 ÷ 3) = 2×E which makes 2 = 2×E Divide both sides of the above equation by 2: 2 ÷ 2 = 2×E ÷ 2 which makes 1 = E making A = 9×E = 9 × 1 = 9 C = 7×E = 7 × 1 = 7 D = 3×E = 3 × 1 = 3 and ABCDEF = 987313