Puzzle for October 19, 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.
Scratchpad
Help Area
Hint #1
Add E to both sides of eq.4: E + F + E = A – E + E which becomes 2×E + F = A In eq.2, replace A with 2×E + F: C + F = 2×E + F Subtract F from each side of the above equation: C + F – F = 2×E + F – F which makes C = 2×E
Hint #2
In eq.3, replace C with 2×E: D – E = 2×E + E Add E to both sides of the equation above: D – E + E = 2×E + E + E which makes D = 4×E
Hint #3
In eq.6, substitute 4×E for D, and 2×E for C: F – 4×E = 2×E – E – F which becomes F – 4×E = E – F Add 4×E and F to both sides of the equation above: F – 4×E + 4×E + F = E – F + 4×E + F which becomes 2×F = 5×E Divide both sides by 2: 2×F ÷ 2 = 5×E ÷ 2 which makes F = 2½×E
Hint #4
Substitute 2½×E for F in eq.4: E + 2½×E = A – E which becomes 3½×E = A – E Add E to both sides of the above equation: 3½×E + E = A – E + E which makes 4½×E = A
Hint #5
Substitute 4½×E for A, 2½×E for F, and 4×E for D in eq.5: 4½×E + B – 2½×E = 4×E + 2½×E which becomes 2×E + B = 6½×E Subtract 2×E from each side of the above equation: 2×E + B – 2×E = 6½×E – 2×E which makes B = 4½×E
Solution
Substitute 4½×E for A and B, 2×E for C, 4×E for D, and 2½×E for F in eq.1: 4½×E + 4½×E + 2×E + 4×E + E + 2½×E = 37 which simplifies to 18½×E = 37 Divide both sides of the above equation by 18½: 18½×E ÷ 18½ = 37 ÷ 18½ which means E = 2 making A = B = 4½×E = 4½ × 2 = 9 C = 2×E = 2 × 2 = 4 D = 4×E = 4 × 2 = 8 F = 2½×E = 2½ × 2 = 5 and ABCDEF = 994825