Question

How many multiples of 18 are there that are less than 3500 and also 2 more than the square of a natural number?

a.6

b.7

c.5

d.8?

Answer 1

Answer:

Multiples of 18 are there that are less than 3500 and also 2 more than the square of a natural number Is 7

Step-by-step explanation:

18k = n²+n : we must have :

n² ≡-2≡16 (mod 18)

⇔(n² ≡0 (mod2);n² ≡7(mod9))

⇔(n ≡ - (mod2); n = {4,5} (mod 9 ))

⇔ n ≡{4,14} (mod18)

since ,

n²+2<3500 ⇔n²< 3498 ⇔n <√3498 ≅59.14

⇒n = { 4, 22, 40, 58 }∪ {14,32,50}

= {4, 14,22, 32, 40 , 50 , 58 }

for each n such that a unique k : |k| = |n | ⇒ the number of positive multiples of 18 :

satisfying the given property = |n| = 7

Option b) is correct