Puzzle for June 18, 2022 ( )
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.
* EF is a 2-digit number (not E×F).
Scratchpad
Help Area
Hint #1
eq.5 may be written as: 10×E + F – A – F = A × E which becomes 10×E – A = A × E In the above equation, add A to both sides, and subtract (A × E) from both sides: 10×E – A + A – (A × E) = A × E + A – (A × E) which becomes 10×E – (A × E) = A which may be written as eq.5a) E × (10 – A) = A
Hint #2
Since A ≠ 10 (A is a one-digit integer), then: 10 – A ≠ 0 Divide both sides of eq.5a by (10 – A): E × (10 – A) ÷ (10 – A) = A ÷ (10 – A) which becomes eq.5b) E = A ÷ (10 – A)
Hint #3
Multiply both sides of eq.3 by A: D × A = (E ÷ A) × A which becomes eq.3a) D × A = E In eq.5b, replace E with D × A (from eq.3a): D × A = A ÷ (10 – A) Since A ≠ 0 (from eq.3), divide both sides by A: (D × A) ÷ A = (A ÷ (10 – A)) ÷ A which becomes eq.5c) D = 1 ÷ (10 – A)
Hint #4
To make eq.5c true, check several possible values for A and D: If A = 9, then D = 1 ÷ (10 – 9) = 1 ÷ 1 = 1 If A = 8, then D = 1 ÷ (10 – 8) = 1 ÷ 2 = ½ If A = 7, then D = 1 ÷ (10 – 7) = 1 ÷ 3 = ⅓ If A < 7, then 0 < D < ⅓ Since D must be a non-negative integer, the above equations make: A = 9 and D = 1
Hint #5
In eq.3a, substitute 1 for D, and 9 for A: 1 × 9 = E which makes 9 = E
Hint #6
Substitute 9 for E and A, and 1 for D in eq.1: 9 + F = 9 + 1 Subtract 9 from each side of the equation above: 9 + F – 9 = 9 + 1 – 9 which makes F = 1
Hint #7
Substitute 1 for D and F in eq.2: C – B = B + 1 + 1 which becomes C – B = B + 2 Add B to both sides of the above equation: C – B + B = B + 2 + B which becomes eq.2a) C = 2×B + 2
Hint #8
Substitute 9 for A, 1 for F and D, and (2×B + 2) for C (from eq.2a) in eq.4: 9 – 1 = (B + 1) × (2×B + 2) which becomes 8 = (B + 1) × (2×B + 2) which may be written as 8 = (B + 1) × 2 × (B + 1) Divide both sides of the above equation by 2: 8 ÷ 2 = (B + 1) × 2 × (B + 1) ÷ 2 which becomes 4 = (B + 1) × (B + 1) which is the same as 4 = (B + 1)² Take the square root of both sides: sq.rt.[4] = sq.rt.[(B + 1)²] which becomes eq.4a) sq.rt.[4] = B + 1
Hint #9
Since B is non-negative, then: B + 1 > 0 Therefore, in eq.4a: sq.rt.[4] > 0 which makes sq.rt.[4] = 2 Substitute 2 for sq.rt.[4] in eq.4a: 2 = B + 1 Subtract 1 from each side of the above equation: 2 – 1 = B + 1 – 1 which makes 1 = B
Solution
Substitute 1 for B in eq.2a: C = 2×1 + 2 which becomes C = 2 + 2 which makes C = 4 and makes ABCDEF = 914191