Two positive integers m and n satisfy \gcd(m, n) = 20 and \text{lcm}(m, n) = 4{}200. It is… — GRE General Test

Two positive integers $m$ and $n$ satisfy $\gcd(m, n) = 20$ and $\text{lcm}(m, n) = 4{}200$ . It is given that $m$ is divisible by $7$ and has exactly $24$ positive integer divisors. If $n$ is a multiple of $3$ , what is the value of $n$ ?

$60$ Cevap
B
$200$
C
$300$
D
$420$
E
$600$

Cevap

The value of

n

60

Prime factorization reveals that

\gcd(m,n) = 2^2 \cdot 3^0 \cdot 5^1 \cdot 7^0

and

\text{lcm}(m,n) = 2^3 \cdot 3^1 \cdot 5^2 \cdot 7^1

. The condition that

n

is a multiple of

3

fixes the exponent of

3

n

1

(so

m

has exponent

0

). The condition that

m

is a multiple of

7

fixes the exponent of

7

m

1

(so

n

has exponent

0

). For

m = 2^a \cdot 3^0 \cdot 5^c \cdot 7^1

, its divisor count equation

2(a+1)(c+1) = 24

simplifies to

(a+1)(c+1) = 12

. Since

a \in \{2, 3\}

and

c \in \{1, 2\}

, the only valid solution is

a=3

and

c=2

. This leaves

n

with exponents

2

for prime

2

1

for prime

3

1

for prime

5

, and

0

for prime

7

, giving

n = 2^2 \cdot 3^1 \cdot 5^1 \cdot 7^0 = 60

Adım Adım Çözüm

Express the GCD and LCM in prime factorized form.

\gcd(m, n) = 20 = 2^2 \cdot 3^0 \cdot 5^1 \cdot 7^0

and

\text{lcm}(m, n) = 4{}200 = 2^3 \cdot 3^1 \cdot 5^2 \cdot 7^1

The prime factor exponents of

m

and

n

must have minimums equal to the GCD exponents and maximums equal to the LCM exponents.

Determine the exponents of prime factors

3

and

7

for

m

and

n

Since

n

is divisible by

3

, exponent of

3

n

1

, so exponent of

3

m

0

. Since

m

is divisible by

7

, exponent of

7

m

1

, so exponent of

7

n

0

Each prime exponent in

\text{lcm}(m, n)

must belong to at least one of the numbers.

Use the divisor count of

m

to find its remaining exponents for primes

2

and

5

Let

m = 2^a \cdot 3^0 \cdot 5^c \cdot 7^1

, where

a \in \{2, 3\}

and

c \in \{1, 2\}

. The number of positive divisors is

(a+1)(0+1)(c+1)(1+1) = 2(a+1)(c+1) = 24

, giving

(a+1)(c+1) = 12

. Testing

a=2 \implies c=3

(invalid range). Testing

a=3 \implies c=2

(valid). Thus

m = 2^3 \cdot 3^0 \cdot 5^2 \cdot 7^1 = 1{}400

The total number of positive integer divisors of

p_1^{e_1} p_2^{e_2} \cdots

is given by

(e_1+1)(e_2+1)\cdots

Determine the prime exponents for

n

and calculate its value.

Since

a=3

n

gets

2^2

. Since

c=2

n

gets

5^1

. Together with

3^1

and

7^0

n = 2^2 \cdot 3^1 \cdot 5^1 \cdot 7^0 = 60

For each prime

p

, the exponent in

n

must match the bound opposite to

m

to satisfy both

\gcd

and

\text{lcm}

Anahtar Kavram

Prime exponent analysis of GCD and LCM alongside the divisor count formula

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