A positive integer n has the prime factorization 2^a \times 3^b \times 5^c, where a, b, an… — ACT

A positive integer $n$ has the prime factorization $2^a \times 3^b \times 5^c$ , where $a$ , $b$ , and $c$ are positive integers. The greatest common divisor of $n$ and $840$ is $120$ , and the least common multiple of $n$ and $90$ is $1,800$ . If $n$ is not divisible by $9$ , what is the value of $a + b + c$ ?

A
5
6Cevap
C
7
D
8
E
9

Cevap

The correct answer is 6. By analyzing the prime factorizations, the greatest common divisor condition

\text{GCD}(n, 840) = 120

tells us that the exponents of the prime factors of

n

satisfy

a \geq 3

b \geq 1

, and

c \geq 1

. The least common multiple condition

\text{LCM}(n, 90) = 1,800

tells us that

a = 3

b \leq 2

, and

c = 2

. Finally, the condition that

n

is not divisible by 9 requires

b < 2

, which forces

b = 1

. Summing these values gives

3 + 1 + 2 = 6

Adım Adım Çözüm

Find the prime factorizations of the given numbers.

840 = 2^3 \times 3^1 \times 5^1 \times 7^1

120 = 2^3 \times 3^1 \times 5^1

90 = 2^1 \times 3^2 \times 5^1

, and

1,800 = 2^3 \times 3^2 \times 5^2

Decomposing the numbers into their prime factors allows us to apply the rules of Greatest Common Divisor (GCD) and Least Common Multiple (LCM) on the exponents.

Analyze the GCD condition.

Since

\text{GCD}(n, 840) = 120

, we have

\min(a, 3) = 3

\min(b, 1) = 1

, and

\min(c, 1) = 1

. This implies

a \geq 3

b \geq 1

, and

c \geq 1

The GCD of two numbers takes the minimum exponent for each prime factor present in both numbers.

Analyze the LCM condition.

Since

\text{LCM}(n, 90) = 1,800

, we have

\max(a, 1) = 3

\max(b, 2) = 2

, and

\max(c, 1) = 2

. This implies

a = 3

b \leq 2

, and

c = 2

The LCM of two numbers takes the maximum exponent for each prime factor present in either number.

Apply the divisibility constraint to determine the exponents.

From previous steps,

a = 3

and

c = 2

. The exponent

b

must satisfy

1 \leq b \leq 2

. Since

n

is not divisible by

9 = 3^2

, the exponent of 3 in the factorization of

n

must be strictly less than 2. Thus,

b = 1

b = 2

, then

n

would contain

3^2

, making it divisible by 9, which violates the given constraint.

Calculate the sum

a + b + c

a + b + c = 3 + 1 + 2 = 6

Adding the determined exponents yields the final requested value.

Anahtar Kavram

Using prime factorizations to find the greatest common divisor and least common multiple of integers.

Alternatif Yöntem

Instead of checking each prime factor independently, we can find the value of

n

directly. Since

\text{GCD}(n, 840) = 120

n

must be a multiple of

120

. The multiples of

120

are

120, 240, 360, 480, 600, 720, 840, \dots

. We test which of these multiples satisfies

\text{LCM}(n, 90) = 1,800

and is not divisible by

9

. Since

1,800 = 15 \times 120

n

must divide

1,800

. The divisors of

1,800

that are multiples of

120

are

120, 240, 360, 600, 900, 1,800

. Among these, only

600

has a greatest common divisor of

120

with

840

, is not divisible by

9

, and has

\text{LCM}(600, 90) = 1,800

. Thus,

n = 600 = 2^3 \times 3^1 \times 5^2

, which gives

a+b+c = 3+1+2 = 6

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