As for two integers, calculation by < , > is defined as the following [Example].
[Example]
< 3, 4 > = 1 / 3×(3+4) =1 / 3×7 = 1/21
< 5, 3 > =1 / 5×(5+3) =1 / 5×8= 1/40
(1) Execute the following calculation in accordance with the rule of a [Example].
① < 8,3 > + < 3,8 >
② < 3,6 > + < 7,2 > + < 6,3 > + < 5,2 >
(2) Find all Integer A which is applicable to the formula,
< A,B > = 1 / 72.
< 3, 4 > = 1 / 3×(3+4) =1 / 3×7 = 1/21
< 5, 3 > =1 / 5×(5+3) =1 / 5×8= 1/40
(1) Execute the following calculation in accordance with the rule of a [Example].
① < 8,3 > + < 3,8 >
② < 3,6 > + < 7,2 > + < 6,3 > + < 5,2 >
(2) Find all Integer A which is applicable to the formula,
< A,B > = 1 / 72.
Answer
Solution
(1) ① It calculates correctly as a rule.
< 8,3 > + < 3,8 > = 1 / 8×(8+3) + 1 / 3×(3+8) = 11 / 8×3×11 = 1 / 8×3 = 1/24.
According to the result of ①, it turns out that
< 3,6 > + < 6,3 > =1 / 3×6 = 1/18.
Therefore,
< 3,6 > + < 7,2 > + < 6,3 > + < 5,2 >
=1 / 18 + 1 / 7×9 + 1 / 5×7
= 1 / 2×9 + 1 / 7×9 + 1 / 5×7
= 35+10+18 / 2×9×5×7
= 63 / 2×9×5×7= 1 / 2×5 = 1/10.
(2) < A,B > = 1 / A x (A+B) = 1 / 72.
Integer A which is applicable to the formula A × (A + B) = 72 is a divisor of 72.
Divisors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
Then A should be applicable to the formula A × (A + B), A is 1, 2, 3, 4, 6, 8.
(1) ① 1/24
② 1 /10
② 1 /10
(2) 1,2,3,4,6,8
(1) ① It calculates correctly as a rule.
< 8,3 > + < 3,8 > = 1 / 8×(8+3) + 1 / 3×(3+8) = 11 / 8×3×11 = 1 / 8×3 = 1/24.
According to the result of ①, it turns out that
< 3,6 > + < 6,3 > =1 / 3×6 = 1/18.
Therefore,
< 3,6 > + < 7,2 > + < 6,3 > + < 5,2 >
=1 / 18 + 1 / 7×9 + 1 / 5×7
= 1 / 2×9 + 1 / 7×9 + 1 / 5×7
= 35+10+18 / 2×9×5×7
= 63 / 2×9×5×7= 1 / 2×5 = 1/10.
(2) < A,B > = 1 / A x (A+B) = 1 / 72.
Integer A which is applicable to the formula A × (A + B) = 72 is a divisor of 72.
Divisors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
Then A should be applicable to the formula A × (A + B), A is 1, 2, 3, 4, 6, 8.
