Raina must choose a number between 55 and 101 that is a multiple of 2, 8 and 10. Write all the numbers that she could choose. If there is more than one number, separate them with commas.

Respuesta :

Answer:

The only possible number is [tex]80[/tex].

Step-by-step explanation:

The number in question needs to be a multiple of all three of [tex]2[/tex], [tex]8[/tex], and [tex]10[/tex]. As a result, it must also be a multiple of the least common multiplier (lcm) of the three number.

Start by finding the least common multiplier of the three numbers.

Factor each number into its prime components:

  • [tex]2[/tex] is a prime number itself.
  • [tex]8 = 2^3[/tex].
  • [tex]10 = 2 \times 5[/tex].

The only prime factors are [tex]2[/tex] and [tex]5[/tex].

  • The greatest power of [tex]2[/tex] among the three numbers is [tex]3[/tex].
  • The greatest power of [tex]5[/tex] among the three numbers is [tex]1[/tex].

Therefore, the least common multiplier of the three number should be the product of [tex]2^3[/tex] and [tex]5[/tex]. That's equal to [tex]2^3 \times 5 = 8 \times 5 = 40[/tex].

In other words, the number (or numbers) in question could be written in the form [tex]40\, k[/tex], where [tex]k[/tex] is an integer.

The question requires that this number be between [tex]55[/tex] and [tex]101[/tex]. In other words,

[tex]55 \le 40\, k \le 101[/tex].

The goal is to find the possible values of [tex]k[/tex]. Note that from integer division by [tex]40[/tex],

  • [tex]55 = 1 \times 40 + 15[/tex], and
  • [tex]101 = 2 \times 40 + 21[/tex].

The inequality becomes:

[tex]1 \times 40 + 15 \le 40\, k \le 2 \times 40 + 21[/tex].

However,

  • [tex]1 \times 40 < 55 = 1 \times 40 + 15[/tex], and
  • [tex]2 \times 40 + 21 = 101 < 3 \times 40[/tex].

Hence,

[tex]1 \times 40 < 1\times 40 + 15 \le 40\, k \le 2 \times 40 + 21 < 3 \times 40[/tex].

[tex]1 \times 40 < 40\, k < 3 \times 40[/tex].

Divide by the positive number [tex]40[/tex] to obtain:

[tex]1 < k < 3[/tex].

Since [tex]k[/tex] is an integers, [tex]k = 2[/tex].

Indeed, [tex]40 \, k = 80[/tex] is between [tex]55[/tex] and [tex]101[/tex].

Therefore, [tex]80[/tex] is the number in question.

Answer:

80

Step-by-step explanation:

80 is a multiple of 2, 8, and 10

*PLz mark brainlies if this helps*

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