Puzzle for December 9, 2019 ( )
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.
Scratchpad
Help Area
Hint #1
In eq.4, replace D with C + E (from eq.2): C + E + E = C + F which becomes C + 2×E = C + F Subtract C from both sides of the above equation: C + 2×E – C = C + F – C which makes 2×E = F
Hint #2
In eq.6, replace F with 2×E: E + 2×E = C which makes 3×E = C
Hint #3
In eq.4, substitute 3×E for C, and 2×E for F: D + E = 3×E + 2×E which becomes D + E = 5×E Subtract E from each side of the equation above: D + E – E = 5×E – E which makes D = 4×E
Hint #4
Substitute 3×E for C, and 4×E for D in eq.5: A = 3×E + 4×E + E which makes A = 8×E
Hint #5
Substitute 2×E for F, 3×E for C, and 8×E for A in eq.3: 2×E – 3×E = 8×E – B Subtract 8×E from both sides of the equation above: 2×E – 3×E – 8×E = 8×E – B – 8×E which means –9×E = –B Multiply both sides by (–1): –9×E × (–1) = –B × (–1) which makes 9×E = B
Solution
Substitute 8×E for A, 9×E for B, 3×E for C, 4×E for D, and 2×E for F in eq.1: 8×E + 9×E + 3×E + 4×E + E + 2×E = 27 which simplifies to 27×E = 27 Divide both sides of the equation above by 27: 27×E ÷ 27 = 27 ÷ 27 which means E = 1 making A = 8×E = 8×1 = 8 B = 9×E = 9×1 = 9 C = 3×E = 3×1 = 3 D = 4×E = 4×1 = 4 F = 2×E = 2×1 = 2 and ABCDEF = 893412