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Question-226697

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Question Number 226697 by Linton last updated on 10/Dec/25
Commented by Linton last updated on 10/Dec/25
The letters in TWO UV PAIRS have the values 0 1 2... 9 in some order with each letter represent ing a different digit.
$${The}\:{letters}\:{in}\:{TWO}\:{UV}\:{PAIRS}\: \\ $$$${have}\:{the}\:{values}\:\mathrm{0}\:\mathrm{1}\:\mathrm{2}…\:\mathrm{9}\:{in}\:{some} \\ $$$${order}\:{with}\:{each}\:{letter}\:{represent} \\ $$$${ing}\:{a}\:{different}\:{digit}. \\ $$
Answered by Raphael254 last updated on 10/Dec/25
UV+UV+V=VAR RP^2 =AIR 2SO = VOW R has the same final digit that 3V and when multiplied by a perfect square the resultant number has the same final digit that R SO ≥ 50 UV is at maximum 99, but 99 + 99 + 9 = 207, and 9 ≠ 2, 98+98+8 = 204, and 8 ≠ 2 and 97+97+7 = 201, and 7 ≠ 2 so V = 1 If V = 1, and R has the same digit that 3V, so R = 3 If RP^2 =AIR, and P ≥6, because AIR has 3 digits, so P = 9, because, P = 6 ⇒ 3×36 = 108, 8≠3 X P=7 ⇒ 3×49 = 147, 7 ≠ 3 X P=8 ⇒ 3×64 = 192, 2≠3 X P=9 ⇒ 3×81 = 243, 3 = 3 ✓ So, if P = 9, AIR = 243, so, A = 2, I = 4 and R = 3 If A = 2, so UV+UV+V=123, and U = 6, because 61+61+1 = 123 For the last, 2SO is a even number, so W is even and SO ≥ 50, from here we could have know that V = 1 too, but, if O = 1, then S1 + S1 = 11W, it is impossible, if O = 2, then S2 + S2 = 12W, it is possible if O = 3, then S3+S3 = 13W, it is impossible if O = 4, then S4+S4 = 14W, it is possible if O = 5, then S5+S5 = 15W, it is possible if O = 6, then S6+S6 = 16W, it is impossible if O = 7, then S7+S7 = 17W, it is possible if O = 8, then S8+S8 = 18W, it is impossible if O = 9, then S9+S9 = 19W, it is possible There are 5 possible cases: where O = 2, S = 6 and W = 4 X where O = 4, S = 7 and W = 8 X where O = 5, S = 7 and W = 0 ✓ where O = 7, S = 8 and W = 4 X where O = 9, S = 9 and W = 8 X The correct is the third line, because all the other repeat numbers, and third line is the only one that appears a 5 that is a number that is missing, so we have: V = 1 A = 2 R = 3 I = 4 O = 5 U = 6 S = 7 and P = 9, only missing 8, but as you noticed, 1234567 = VARIOUS To confirm: 61+61+1 = 123 3×9×9 = 243 75+75=150
$$ \\ $$$${UV}+{UV}+{V}={VAR} \\ $$$${RP}^{\mathrm{2}} ={AIR} \\ $$$$\mathrm{2}{SO}\:=\:{VOW} \\ $$$$ \\ $$$${R}\:{has}\:{the}\:{same}\:{final}\:{digit}\:{that}\:\mathrm{3}{V}\:{and}\:{when} \\ $$$${multiplied}\:{by}\:{a}\:{perfect}\:{square}\:{the}\:{resultant} \\ $$$${number}\:{has}\:{the}\:{same}\:{final}\:{digit}\:{that}\:{R} \\ $$$$ \\ $$$${SO}\:\geqslant\:\mathrm{50} \\ $$$$ \\ $$$${UV}\:{is}\:{at}\:{maximum}\:\mathrm{99}, \\ $$$${but}\:\mathrm{99}\:+\:\mathrm{99}\:+\:\mathrm{9}\:=\:\mathrm{207},\:{and}\:\mathrm{9}\:\neq\:\mathrm{2}, \\ $$$$\mathrm{98}+\mathrm{98}+\mathrm{8}\:=\:\mathrm{204},\:{and}\:\mathrm{8}\:\neq\:\mathrm{2} \\ $$$${and}\:\mathrm{97}+\mathrm{97}+\mathrm{7}\:=\:\mathrm{201},\:{and}\:\mathrm{7}\:\neq\:\mathrm{2} \\ $$$${so}\:{V}\:=\:\mathrm{1} \\ $$$$ \\ $$$${If}\:{V}\:=\:\mathrm{1},\:{and}\:{R}\:{has}\:{the}\:{same}\:{digit} \\ $$$${that}\:\mathrm{3}{V},\:{so}\:{R}\:=\:\mathrm{3} \\ $$$$ \\ $$$${If}\:{RP}^{\mathrm{2}} ={AIR},\:{and}\:{P}\:\geqslant\mathrm{6},\:{because}\:{AIR}\:{has} \\ $$$$\mathrm{3}\:{digits},\:{so}\:{P}\:=\:\mathrm{9},\:{because}, \\ $$$$ \\ $$$${P}\:=\:\mathrm{6}\:\Rightarrow\:\mathrm{3}×\mathrm{36}\:=\:\mathrm{108},\:\mathrm{8}\neq\mathrm{3}\:{X} \\ $$$${P}=\mathrm{7}\:\Rightarrow\:\mathrm{3}×\mathrm{49}\:=\:\mathrm{147},\:\mathrm{7}\:\neq\:\mathrm{3}\:{X} \\ $$$${P}=\mathrm{8}\:\Rightarrow\:\mathrm{3}×\mathrm{64}\:=\:\mathrm{192},\:\mathrm{2}\neq\mathrm{3}\:{X} \\ $$$${P}=\mathrm{9}\:\Rightarrow\:\mathrm{3}×\mathrm{81}\:=\:\mathrm{243},\:\mathrm{3}\:=\:\mathrm{3}\:\checkmark \\ $$$$ \\ $$$${So},\:{if}\:{P}\:=\:\mathrm{9},\:{AIR}\:=\:\mathrm{243},\:{so}, \\ $$$${A}\:=\:\mathrm{2},\:{I}\:=\:\mathrm{4}\:{and}\:{R}\:=\:\mathrm{3} \\ $$$$ \\ $$$${If}\:{A}\:=\:\mathrm{2},\:{so}\:{UV}+{UV}+{V}=\mathrm{123},\:{and} \\ $$$${U}\:=\:\mathrm{6},\:{because}\:\mathrm{61}+\mathrm{61}+\mathrm{1}\:=\:\mathrm{123} \\ $$$$ \\ $$$${For}\:{the}\:{last},\:\mathrm{2}{SO}\:{is}\:{a}\:{even}\:{number}, \\ $$$${so}\:{W}\:{is}\:{even}\:{and}\:{SO}\:\geqslant\:\mathrm{50},\:{from}\:{here}\:{we}\:{could} \\ $$$${have}\:{know}\:{that}\:{V}\:=\:\mathrm{1}\:{too},\:{but}, \\ $$$$ \\ $$$${if}\:{O}\:=\:\mathrm{1},\:{then}\:{S}\mathrm{1}\:+\:{S}\mathrm{1}\:=\:\mathrm{11}{W},\:{it}\:{is}\:{impossible}, \\ $$$${if}\:{O}\:=\:\mathrm{2},\:{then}\:{S}\mathrm{2}\:+\:{S}\mathrm{2}\:=\:\mathrm{12}{W},\:{it}\:{is}\:{possible} \\ $$$${if}\:{O}\:=\:\mathrm{3},\:{then}\:{S}\mathrm{3}+{S}\mathrm{3}\:=\:\mathrm{13}{W},\:{it}\:{is}\:{impossible} \\ $$$${if}\:{O}\:=\:\mathrm{4},\:{then}\:{S}\mathrm{4}+{S}\mathrm{4}\:=\:\mathrm{14}{W},\:{it}\:{is}\:{possible} \\ $$$${if}\:{O}\:=\:\mathrm{5},\:{then}\:{S}\mathrm{5}+{S}\mathrm{5}\:=\:\mathrm{15}{W},\:{it}\:{is}\:{possible} \\ $$$${if}\:{O}\:=\:\mathrm{6},\:{then}\:{S}\mathrm{6}+{S}\mathrm{6}\:=\:\mathrm{16}{W},\:{it}\:{is}\:{impossible} \\ $$$${if}\:{O}\:=\:\mathrm{7},\:{then}\:{S}\mathrm{7}+{S}\mathrm{7}\:=\:\mathrm{17}{W},\:{it}\:{is}\:{possible} \\ $$$${if}\:{O}\:=\:\mathrm{8},\:{then}\:{S}\mathrm{8}+{S}\mathrm{8}\:=\:\mathrm{18}{W},\:{it}\:{is}\:{impossible} \\ $$$${if}\:{O}\:=\:\mathrm{9},\:{then}\:{S}\mathrm{9}+{S}\mathrm{9}\:=\:\mathrm{19}{W},\:{it}\:{is}\:{possible} \\ $$$$ \\ $$$${There}\:{are}\:\mathrm{5}\:{possible}\:{cases}: \\ $$$$ \\ $$$${where}\:{O}\:=\:\mathrm{2},\:{S}\:=\:\mathrm{6}\:{and}\:{W}\:=\:\mathrm{4}\:{X} \\ $$$${where}\:{O}\:=\:\mathrm{4},\:{S}\:=\:\mathrm{7}\:{and}\:{W}\:=\:\mathrm{8}\:{X} \\ $$$${where}\:{O}\:=\:\mathrm{5},\:{S}\:=\:\mathrm{7}\:{and}\:{W}\:=\:\mathrm{0}\:\checkmark \\ $$$${where}\:{O}\:=\:\mathrm{7},\:{S}\:=\:\mathrm{8}\:{and}\:{W}\:=\:\mathrm{4}\:{X} \\ $$$${where}\:{O}\:=\:\mathrm{9},\:{S}\:=\:\mathrm{9}\:{and}\:{W}\:=\:\mathrm{8}\:{X} \\ $$$$ \\ $$$${The}\:{correct}\:{is}\:{the}\:{third}\:{line},\:{because} \\ $$$${all}\:{the}\:{other}\:{repeat}\:{numbers}, \\ $$$${and}\:{third}\:{line}\:{is}\:{the}\:{only}\:{one}\:{that} \\ $$$${appears}\:{a}\:\mathrm{5}\:{that}\:{is}\:{a}\:{number}\:{that}\:{is}\:{missing}, \\ $$$${so}\:{we}\:{have}: \\ $$$$ \\ $$$${V}\:=\:\mathrm{1} \\ $$$${A}\:=\:\mathrm{2} \\ $$$${R}\:=\:\mathrm{3} \\ $$$${I}\:=\:\mathrm{4} \\ $$$${O}\:=\:\mathrm{5} \\ $$$${U}\:=\:\mathrm{6} \\ $$$${S}\:=\:\mathrm{7} \\ $$$$ \\ $$$${and}\:{P}\:=\:\mathrm{9},\:{only}\:{missing}\:\mathrm{8},\:{but}\:{as}\:{you}\:{noticed}, \\ $$$$ \\ $$$$\mathrm{1234567}\:=\:{VARIOUS} \\ $$$$ \\ $$$${To}\:{confirm}: \\ $$$$ \\ $$$$\mathrm{61}+\mathrm{61}+\mathrm{1}\:=\:\mathrm{123} \\ $$$$\mathrm{3}×\mathrm{9}×\mathrm{9}\:=\:\mathrm{243} \\ $$$$\mathrm{75}+\mathrm{75}=\mathrm{150} \\ $$

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