Two positive integers x and y satisfy x < y, \text{GCD}(x, y) = 15, and \text{LCM}(x, y) =… — GMAT

Two positive integers $x$ and $y$ satisfy $x < y$ , $\text{GCD}(x, y) = 15$ , and $\text{LCM}(x, y) = 1,800$ . If $x$ is not a multiple of $9$ and $y$ is not a multiple of $25$ , what is the value of $y - x$ ?

Cevap: 285

Cevap

The value of

y - x

is 285.

Expressing

x

and

y

15a

and

15b

yields

ab = 120

. Testing coprime pairs

(a,b)

shows that only

(5, 24)

gives values

x = 75

and

y = 360

that satisfy both constraints (75 is not a multiple of 9, and 360 is not a multiple of 25). The difference is

360 - 75 = 285

Adım Adım Çözüm

Set up coprime factor representation

x = 15a

and

y = 15b

with

\text{GCD}(a, b) = 1

and

a < b

Any two numbers can be expressed as their GCD multiplied by coprime integer quotients.

Solve for the product of coprime quotient factors

a \cdot b

a \cdot b = 120

The LCM of two numbers divided by their GCD equals the product of their coprime quotients.

Identify candidate coprime pairs

(a, b)

(1, 120), (3, 40), (5, 24), (8, 15)

Since

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

, all prime factor powers must be assigned entirely to either

a

b

to maintain

\text{GCD}(a, b) = 1

Filter pairs using divisibility constraints on

x

and

y

Only

(75, 360)

satisfies

x \not\equiv 0 \pmod 9

and

y \not\equiv 0 \pmod{25}

75 = 3 \cdot 25

(not divisible by 9) and

360 = 8 \cdot 9 \cdot 5

(not divisible by 25).

Calculate the target difference

y - x

285

360 - 75 = 285

Anahtar Kavram

Relationship between GCD, LCM, and prime factorization distribution in coprime quotients.

Soru

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