Let a and b be positive integers such that \gcd(a, b) = 60 and \text{lcm}(a, b) = 75,600. … — GRE General Test

Let $a$ and $b$ be positive integers such that $\gcd(a, b) = 60$ and $\text{lcm}(a, b) = 75,600$ . Which of the following statements must be true? Select all that apply.

The product $ab$ is equal to $4,536,000$ .Cevap
Neither $a$ nor $b$ can be divisible by $49$ .Cevap
There are exactly $8$ distinct unordered pairs of positive integers $\{a, b\}$ that satisfy the given conditions.Cevap
D
Both $a$ and $b$ must be multiples of $9$ .
E
The greatest common divisor of $(a + b)$ and $60$ is equal to $30$ .

Cevap

The statements asserting that

ab = 4,536,000

, that neither number is divisible by

49

, and that there are exactly

8

distinct unordered pairs

\{a, b\}

are all correct.

The product of the GCD and LCM of two numbers always yields their product, confirming

ab = 4,536,000

. The prime factorization of the LCM shows that

7

appears only to the first power, making divisibility by

49

impossible for either number. Finally, assigning the minimum and maximum prime exponents across the four distinct prime factors

(2, 3, 5, 7)

yields

2^4 = 16

ordered pairs, which corresponds to exactly

8

unordered pairs.

Adım Adım Çözüm

Apply the product identity for GCD and LCM.

ab = \gcd(a, b) \times \text{lcm}(a, b) = 60 \times 75,600 = 4,536,000

The product of two positive integers is always equal to the product of their greatest common divisor and least common multiple.

Find the prime factorizations of

\gcd(a, b)

and

\text{lcm}(a, b)

to determine exponent bounds.

\gcd(a, b) = 2^2 \times 3^1 \times 5^1 \times 7^0

and

\text{lcm}(a, b) = 2^4 \times 3^3 \times 5^2 \times 7^1

The exponent of each prime factor in

\gcd(a, b)

gives the minimum exponent in

a

and

b

, while the exponent in

\text{lcm}(a, b)

gives the maximum exponent.

Evaluate prime factor constraints for prime

7

and prime

3

Maximum power of

7

a

b

7^1 = 7

, so neither can be divisible by

49

. For prime

3

, one integer has

3^1

(not divisible by

9

) and the other has

3^3 = 27

An integer is divisible by

49 = 7^2

only if its prime factorization contains

7^2

or higher. Similarly, a number is a multiple of

9 = 3^2

only if its prime exponent for

3

is at least

2

Calculate the total number of distinct unordered pairs

\{a, b\}

For each of the

4

prime factors (

2, 3, 5, 7

), there are

2

choices for which integer gets the higher exponent. Total ordered pairs

(a, b) = 2^4 = 16

. Unordered pairs

\{a, b\} = 16 / 2 = 8

Since

\min < \max

for all prime factors,

a \neq b

for all pairs, allowing exact division by

2

to account for symmetry.

Analyze

\gcd(a + b, 60)

Since

60 \mid a

and

60 \mid b

, it follows that

60 \mid (a + b)

, making

\gcd(a + b, 60) = 60

If a number

k

divides both

a

and

b

, it must divide their linear combination

a + b

Anahtar Kavram

Prime Factorization, GCD-LCM Identities, and Counting Valid Integer Pairs

Soru

Cevap

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