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The LCM of two non-zero integers, x(64) and y(80), is the smallest positive integer m(320) that is divisible by both x(64) and y(80) without any remainder.

The LCM of 64 and 80 is 320. To find the LCM (least common multiple) of 64 and 80, we need to find the multiples of 64 and 80 (multiples of 64 = 64, 128, 192, 256 . . . . 320; multiples of 80 = 80, 160, 240, 320) and choose the smallest multiple that is exactly divisible by 64 and 80, i.e., 320.

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To calculate the LCM of 64 and 80 by the division method, we will divide the numbers(64, 80) by their prime factors (preferably common). The product of these divisors gives the LCM of 64 and 80.

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LCM of 64 and 80 is the smallest number among all common multiples of 64 and 80. The first few multiples of 64 and 80 are (64, 128, 192, 256, 320, 384, 448, . . . ) and (80, 160, 240, 320, 400, 480, 560, . . . ) respectively. There are 3 commonly used methods to find LCM of 64 and 80 - by division method, by prime factorization, and by listing multiples.

LCM(80, 64) × GCF(80, 64) = 80 × 64 Since the LCM of 80 and 64 = 320 ⇒ 320 × GCF(80, 64) = 5120 Therefore, the greatest common factor (GCF) = 5120/320 = 16.

The LCM of 64 and 80 is the product of all prime numbers on the left, i.e. LCM(64, 80) by division method = 2 × 2 × 2 × 2 × 2 × 2 × 5 = 320.

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The value of LCM of 64, 80 is the smallest common multiple of 64 and 80. The number satisfying the given condition is 320.

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To find the LCM of 64 and 80 using prime factorization, we will find the prime factors, (64 = 2 × 2 × 2 × 2 × 2 × 2) and (80 = 2 × 2 × 2 × 2 × 5). LCM of 64 and 80 is the product of prime factors raised to their respective highest exponent among the numbers 64 and 80. ⇒ LCM of 64, 80 = 26 × 51 = 320.

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Given: GCD = 16 product of numbers = 5120 ∵ LCM × GCD = product of numbers ⇒ LCM = Product/GCD = 5120/16 Therefore, the LCM is 320. The probable combination for the given case is LCM(64, 80) = 320.

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The relation between GCF and LCM of 64 and 80 is given as, LCM(64, 80) × GCF(64, 80) = Product of 64, 80 Prime factorization of 64 and 80 is given as, 64 = (2 × 2 × 2 × 2 × 2 × 2) = 26 and 80 = (2 × 2 × 2 × 2 × 5) = 24 × 51 LCM(64, 80) = 320 GCF(64, 80) = 16 LHS = LCM(64, 80) × GCF(64, 80) = 320 × 16 = 5120 RHS = Product of 64, 80 = 64 × 80 = 5120 ⇒ LHS = RHS = 5120 Hence, verified.

Prime factorization of 64 and 80 is (2 × 2 × 2 × 2 × 2 × 2) = 26 and (2 × 2 × 2 × 2 × 5) = 24 × 51 respectively. LCM of 64 and 80 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 26 × 51 = 320. Hence, the LCM of 64 and 80 by prime factorization is 320.