19. Find the sum of even factors of 5400.
a. 10360
b. 11360
c. 14360
d. 17360
Answer: D
Explanation:
\(5400\) = \({2^3} \times {3^3} \times {5^2}\)
To find the sum of even factors we should choose atleast one 2 multiple of $2^3$.
Sum of even factors = $\left( {2 + {2^2} + {2^3}} \right)\left( {1 + 3 + {3^2} + {3^3}}\right)\left( {1 + 5 + {5^2}} \right)$
$\therefore$ Sum of even factors = $14 \times 40 \times 31 = 17360$
20. Find the number of factors of 5400 which are multiples of 3.
a. 36
b. 39
c. 40
d. 52
Answer: A
Explanation:
$5400 = {2^3} \times {3^3} \times {5^2}$
To get factors which are multiples of 3, we have to choose atleast one $3$ multiple.
So one of $3,\,{3^2},\,{3^3}$ must be chosen.
Remaining factors of ${2^3} \times {5^2}$ are $\left( {3 + 1} \right).\left( {2 + 1} \right)$
$\therefore$ Total factors which are multiples of 3 = $4 \times 3 \times 3 = 36$
21. Find the number of factors of 5400 which are multiples of 15.
a. 15
b. 16
c. 24
d. 28
Answer: C
Explanation:
$5400 = {2^3} \times {3^3} \times {5^2}$
To get factors which are multiples of 15, we have to choose atleast one $3$ multiple and one $5$ multiple.
So one of $3,\,{3^2},\,{3^3}$ must be chosen and one of $5,\,{5^2}$ must be chosen.
$\therefore$ Total factors which are multiples of 15 = $4 \times 3 \times 2 = 24$
22. Find the sum of factors of 5400 which are multiples of 5.
a. 18450
b. 19200
c. 27200
d. 37200
Answer: D
Explanation:
$5400 = {2^3} \times {3^3} \times {5^2}$
To get sum of factors which are multiples of 5, we have to choose atleast one $5$ multiple.
So one of $5,\,{5^2}$ must be chosen.
$\therefore$ Sum of factors which are multiples of 5 = $\left( {1 + 2 + {2^2} + {2^3} + {2^4}} \right)\left( {1 + 3 + {3^2} + {3^3}} \right)\left({5 + {5^2}} \right)$ $= 37200$
