The sum of two integers is divided by the greatest common divisor of the two.
For example, since the greatest common divisor of 12 and 8 is 4, it is (12 + 8) / 4 = 5.
This will be expressed as 12 # 8 = 5.
Moreover, the product of two integers is divided by the least common multiple of the two.
For example, since the least common multiple of 12 and 8 is 24, it is (12 × 8) / 24 = 4.
This will be expressed as 12 & 8 = 4.
Answer the following questions.
(1) Find the answer to the following calculation .
(1)-1 60 # 84
(1)-2 (24 & 60) # 51
(2) Answer one integer of two digit applicable to P and Q, respectively.
(2)-1 (16 # P) & 14 = 7
(2)-2 (54 & 30) # ( 48 & Q) = 7
This will be expressed as 12 # 8 = 5.
Moreover, the product of two integers is divided by the least common multiple of the two.
For example, since the least common multiple of 12 and 8 is 24, it is (12 × 8) / 24 = 4.
This will be expressed as 12 & 8 = 4.
Answer the following questions.
(1) Find the answer to the following calculation .
(1)-1 60 # 84
(1)-2 (24 & 60) # 51
(2) Answer one integer of two digit applicable to P and Q, respectively.
(2)-1 (16 # P) & 14 = 7
(2)-2 (54 & 30) # ( 48 & Q) = 7
Answer
(1) -2 Since the least common multiple of 24 and 60 is 120,
24 & 60 = (24 x 60) / 120 = 12.
Since the greatest common divisor is 3,
12 # 51 = (12 + 51) / 3 = 21.
(2)-1 The greatest common divisor of A and B is set to G in the next calculation, since A = G × a and B = G × b, A × B = G × G × a × b.
By the way, as G × a × b is the least common multiple of A and B, it turns out that A× B = (greatest common divisor) × (the least common multiple).
In this calculation, x = 1, 3, 5, 7, ----- and C is 7, 21, 35, 49 -----.
(2)-2 54 & 30 = (54 × 30) / 270 = 6.
(1)-1 12
(1)-2 21
(2)-1 one of 12, 19, 26, 33, 40, 47, 54, 61, 68, 75, 82, 89, 96
(2)-2 one of the prime number of two digit or 25, 35, 40, 49, 55, 56, 65, 77, 85, 88, 91, 95
(1)-2 21
(2)-1 one of 12, 19, 26, 33, 40, 47, 54, 61, 68, 75, 82, 89, 96
(2)-2 one of the prime number of two digit or 25, 35, 40, 49, 55, 56, 65, 77, 85, 88, 91, 95
Solution
(1)-1 As the greatest common divisor of 60 and 84 is 12,
60 # 84 = (60 + 84) / 12 = 12
60 # 84 = (60 + 84) / 12 = 12
(1) -2 Since the least common multiple of 24 and 60 is 120,
24 & 60 = (24 x 60) / 120 = 12.
Since the greatest common divisor is 3,
12 # 51 = (12 + 51) / 3 = 21.
(2)-1 The greatest common divisor of A and B is set to G in the next calculation, since A = G × a and B = G × b, A × B = G × G × a × b.
By the way, as G × a × b is the least common multiple of A and B, it turns out that A× B = (greatest common divisor) × (the least common multiple).
At this problem, A # B = ( A × B) / (the least common multiple).
The answer of this calculation is the same as the greatest common divisor of A and B.
Calculation of A # B is to find the greatest common divisor of A and B.
The answer of 16 # P is set to C.
The answer of 16 # P is set to C.
C # 14 = 7 means that the greatest common divisor of C and 14 is 7.
In this calculation, x = 1, 3, 5, 7, ----- and C is 7, 21, 35, 49 -----.
Assuming 16 # P = 7, the divisor of 16 is 1, 2, 4, 8, and 16.
The greatest common divisor of 16 and P may also be 1, 2, 4, 8, or 16.
Check each calculation on by one to confirm which the greatest common divisor is applicable.
In case the greatest common denominator = 1, 16 # P = 7 × 1 = 7 and P is not found.
In case the greatest common denominator = 2, 16 # P = 7 × 2 = 14 and P is not found.
In case the greatest common denominator = 4, 16 # P = 7 × 4 = 28 and P = 28 - 16 =12.
Other answers are 19, 26, 33, 40, 47, 54, 61, 68, 75, 82, 89, and 96.
In case the greatest common denominator = 1, 16 # P = 7 × 1 = 7 and P is not found.
In case the greatest common denominator = 2, 16 # P = 7 × 2 = 14 and P is not found.
In case the greatest common denominator = 4, 16 # P = 7 × 4 = 28 and P = 28 - 16 =12.
Other answers are 19, 26, 33, 40, 47, 54, 61, 68, 75, 82, 89, and 96.
(2)-2 54 & 30 = (54 × 30) / 270 = 6.
(You may ask for the greatest common denominator of 54 and 30).
The answer of the calculation of 48 &Q is set to D, 6 & D = 7.
The answer of the calculation of 48 &Q is set to D, 6 & D = 7.
Since the divisor of 6 is 1, 2, 3, and 6, the greatest common divisor of 6 and D may also be 1, 2, 3, or 6.
In case the greatest common divisor is 1, 6 & D = 7 × 1 = 7 and D = 7 - 6 = 1.
In case the greatest common divisor is 1, 6 & D = 7 × 1 = 7 and D = 7 - 6 = 1.
Then D = 1 means 48 & Q = 1.
This means that the answer of 48 × Q is same as the least common multiple of 48 and Q.
Q which is an integer of two digit turns to be a prime number of two digit. One of them is 11.
Other answers are 25, 35, 40, 49, 55, 56, 65, 77, 85, 88, 91, and 95 in addition to the prime number of two digit.
