Question

Find the remainder when (5m+1)5m+3)(5m+4) is divided by 5 if it is a natural no

Answer 1

Answer:

Remainder is 2

Given:

(5m + 1)(5m + 3)(5m + 4)

To find:

If (5m + 1)(5m + 3)(5m + 4) is divided by 5, what is the remainder.

Solution:

Any integer which is more than zero is otherwise called as a natural number. They are used for counting and so called as counting numbers. They lack decimal points since they are integers but may have commas in large numbers.

\Rightarrow \frac{(5 m+1)(5 m+3)(5 m+4)}{5}⇒

5

(5m+1)(5m+3)(5m+4)

If m = 1, then 5m = 5, so neglect it from the numerator.

\Rightarrow \frac{1 \times 3 \times 4}{5}⇒

5

1×3×4

\Rightarrow \frac{12}{5}⇒

5

12

Then, the Remainder will be 2

Hope you like this

Answer 2

Answer:

2

Explanation:

Easy way... Use congruences:

(5m+1)(5m+3)(5m+4) ≡ (1)(3)(4) = 12 ≡ 2 (mod 5)

Harder way...  Expand:

(5m+1)(5m+3)(5m+4)

= 5³m³ + 5²m²(1+3+4) + 5m((1)(3)+(1)(4)+(3)(4)) + (1)(3)(4)

= 5³m³ + 5²m²×8 + 5m×19 + 12

= 5 × ( 5²m³ + 5m²×8 + m×19 + 2 ) + 2

[ last step used 12 = 5 × 2 + 2 ]

So when (5m+1)(5m+3)(5m+4) is divided by 5, the remainder is 2.