Let ‘n’ be the smallest positive integer having exactly 21 positive integer divisors including 1 and the number itself, then the units digit of ‘n’ is___

Q. Find the smallest positive integer such that it has exactly

100

different positive divisors including 1 and the number itself.

Q. If the set of factors of a whole number

n

including

n

itself but not

1

is denoted by

F (n)

and

F (16) \cap F (40) = F (X)

, then

X

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Let d(n) denote the number of divisors of n including 1 and itself. Then d(225),d(1125) and d(640) are

The correct option is D in GP225=32×52⇒d(225)=3×3=91125=32×53⇒d(1125)=3×4=12640=27×5⇒d(640)=8×2=169,12,16 are in G.P.

Let $d (n)$ denote the number of divisors of $n$ including $1$ and itself. Then $d (225), d (1125)$ and $d (640)$ are

The correct option is D in GP
$225 = 3^{2} \times 5^{2} \Rightarrow d (225) = 3 \times 3 = 9$
$1125 = 3^{2} \times 5^{3} \Rightarrow d (1125) = 3 \times 4 = 12$
$640 = 2^{7} \times 5 \Rightarrow d (640) = 8 \times 2 = 16$
$9, 12, 16$ are in G.P.