A certain number when divided by 45 leaves a remainder of 37. What is the remainder if the same number is divided by 15?

A. 13
B. 11
C. 9
D. 1



Answer :

Sure! Let's solve the problem step by step.

We are given a number that, when divided by 45, leaves a remainder of 37. We need to find the remainder when this same number is divided by 15.

1. Understanding the problem:
- Let's denote this number by [tex]\( n \)[/tex].
- Mathematically, if [tex]\( n \)[/tex] leaves a remainder of 37 when divided by 45, it can be written as:
[tex]\[ n = 45k + 37 \][/tex]
for some integer [tex]\( k \)[/tex].

2. Find the remainder when dividing by 45:
- Since the problem states that [tex]\( n \)[/tex] leaves a remainder of 37 when divided by 45, it means:
[tex]\[ n \equiv 37 \mod 45 \][/tex]
- So, [tex]\( n \)[/tex] must have the form of 45k + 37 for some integer [tex]\( k \)[/tex].

3. Dividing by 15:
- To find the remainder when [tex]\( n \)[/tex] is divided by 15, we can use:
[tex]\[ n \text{ mod } 15 \][/tex]
- Substitute [tex]\( n \)[/tex] with [tex]\( 45k + 37 \)[/tex]:
[tex]\[ (45k + 37) \mod 15 \][/tex]

4. Break down the modulo operation:
- We know that:
[tex]\[ 45 = 3 \times 15 \][/tex]
Thus:
[tex]\[ 45k \equiv 0 \mod 15 \][/tex]
- So we can simplify our expression:
[tex]\[ (45k + 37) \mod 15 \equiv 37 \mod 15 \][/tex]

5. Find the remainder of 37 when divided by 15:
- [tex]\( 37 \div 15 = 2 \)[/tex] with a remainder of 7.

Thus, when we divide the same number by 15, the remainder is 7.

Therefore, the correct answer is:
(d) 7