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

# How many natural numbers n are there such that n!+10 is a perfect square?

A
1
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
B
2
No worries! Weâ€˜ve got your back. Try BYJUâ€˜S free classes today!
C
4
No worries! Weâ€˜ve got your back. Try BYJUâ€˜S free classes today!
D
Infinitely many.
No worries! Weâ€˜ve got your back. Try BYJUâ€˜S free classes today!
Open in App
Solution

## The correct option is B 1If n=1,2,4,5 then n!+10 is not prefect squareIf n=3, n!+10=16 which is a perfect squareFor n>5,n!+10=(1×2×3×4×5×6×....×n)+10Taking 10 commonn!+10=(1×2×3×4×5×6×....×n)+10=10[(1×3×4×6....×n)+1]=2×5[(1×3×4×6....×n)+1]You can clealry see term outside the brackets is even and inside is odd, so the exponent of 2 will always remain 1So, n!+10 can not be a perfect square for n>5So, option A is correct as we got one value of n which is 3.

Suggest Corrections
0
Join BYJU'S Learning Program
Related Videos
Interesting Patterns
MATHEMATICS
Watch in App
Join BYJU'S Learning Program