The greatest integer which divides p -1 p -2 p -3 - p - q for all p - N and fixed q - N isA- p -B- q-C- PD- q

The correct option is B q!(p+1)(p+2)(p+3)...(p+q), ∀ p∈ N Put p=1 (p+1)(p+2)(p+3)...(p+q) =2×3 × 4 ×⋯× (q+1) =(q+1)!1! Similarly, for p=2 (p+1)(p+2)(p+3 ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

AM,GM,HM Inequality

The greatest ...

Question

The greatest integer which divides $(p + 1) (p + 2) (p + 3) . . . (p + q)$ for all $p \in N$ and fixed $q \in N$ is

p!

No worries! We‘ve got your back. Try BYJU‘S free classes today!

q!

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

p

No worries! We‘ve got your back. Try BYJU‘S free classes today!

q

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B $q!$
$(p + 1) (p + 2) (p + 3) . . . (p + q), \forall p \in N$
Put $p = 1$
$(p + 1) (p + 2) (p + 3) . . . (p + q) = 2 \times 3 \times 4 \times \dots \times (q + 1) = \frac{(q + 1)!}{1!}$

Similarly, for $p = 2$
$(p + 1) (p + 2) (p + 3) . . . (p + q)$
$= \frac{(q + 2)!}{2!}$

$∴ (p + 1) (p + 2) (p + 3) . . . (p + q) = \frac{q! \times (p + q)!}{p! q!} = q! \times^{p + q} C_{p}$
So, it will be divisible by $q!$

Alternative:
In general, the product of $r$ consecutive integers is divisible by $r!$
$∴ (p + 1) (p + 2) (p + 3) . . . (p + q)$ is divisible by $q!$

Suggest Corrections

Similar questions

Q. The greatest integer which divides

(p + 1) (p + 2) (p + 3) . . . . (p + q)

for all

p ϵ N

and fixed

q ϵ N

What is the angle between $\to P$ and the resultant of $(\to P + \to Q)$ and $(\to P - \to Q)$

Q. If two positive integers m and n are expressible in the form m = pq³ and n = p³q², where p, q are prime numbers, then HCF (m, n) =
(a) pq
(b) pq2
(c) p³q²
(d) p²q²

Q. The logically equivalent proposition of

p \Leftrightarrow q

The logically equivalent proposition of $p \Leftrightarrow q$ is

Join BYJU'S Learning Program

The greatest integer which divides (p+1)(p+2)(p+3)...(p+q) for all p∈N and fixed q∈N is

The greatest integer which divides $(p + 1) (p + 2) (p + 3) . . . (p + q)$ for all $p \in N$ and fixed $q \in N$ is