How do you find a and b?

Question

How do you find a and b?

i) a is such that has 12 divisors
ii) b has 18 divisors and
iii) gcd (a,b)=45

2 Answers

Impact of this question

3909 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-02-26T19:41:08

$a = 90$ $a=90$
$b = 171$ $b=171$

Explanation:

$a = 45 x_{1}$ $a=45x_1$

$b = 45 x_{2}$ $b=45x_2$

Divisors of $45 \to 1 \times 45, d 3 \times 15, d 5 \times 9$ $45 -> 1xx45, color(white)("d")3xx15,color(white)("d")5xx9$ count of 6

Given that $a \to 12$ $a->12$ divisors then $2 \times 45 \to 2 \times 6 divisors$ $2xx45->2xx"6 divisors"$

Given that $b \to 18$ $b->18$ divisors then $3 \times 45 \to 3 \times 6 divisors$ $3xx45->3xx"6 divisors"$

$3 \times 45 = 171$ $3xx45= 171$
$2 \times 45 = 90$ $2xx45=90$

Tony B

Answer 2 · 2018-03-01T12:17:02

Another approach.

Explanation:

Given a number $N = n \prod k = 1 p_{k}^{α_{k}}$ $N = prod_(k=1)^n p_k^(alpha_k)$

with $p_{k}$ $p_k$ being the primes and $α_{k}$ $alpha_k$ their multiplicity we have

$n \prod k = 1 (α_{k} + 1)$ $prod_(k=1)^n (alpha_k+1)$ divisors for $N$ $N$

Assuming now that $a$ $a$ and $b$ $b$ are composed by the same prime factors $3$ $3$ and $5$ $5$ we have

$a = 3^{2} \times 5_{1}^{α_{1}}$ $a = 3^2 xx 5^(alpha_1)$
$b = 3_{2}^{α_{2}} \times 5$ $b = 3^(alpha_2) xx 5$

The number of factors for $a$ $a$ and $b$ $b$ are

${(a->(2+1)(alpha_1+1)= 12),(b->(alpha_2+1)(1+1)=18):}$

now solving for $alpha_1, alpha_2$ we have

$alpha_1 = 3$ and $alpha_2 = 8$

then

$a = 3^2 xx 5^3$ and
$b = 3^8 xx 5$

$a$ factors are

${1, 3, 5, 9, 15, 25, 45, 75, 125, 225, 375, 1125}$

$b$ factors are

${1, 3, 5, 9, 15, 27, 45, 81, 135, 243, 405, 729, 1215, 2187, 3645, 6561, 10935, 32805}$

Science

Math

Humanities

... and beyond

How do you find a and b?

i) a is such that has 12 divisors
ii) b has 18 divisors and
iii) gcd (a,b)=45

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you find a and b?

i) a is such that has 12 divisors ii) b has 18 divisors and iii) gcd (a,b)=45

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question

i) a is such that has 12 divisors
ii) b has 18 divisors and
iii) gcd (a,b)=45