Question

Find the least number which when divided by \[12\], leaves a remainder of \[7\], when divided by \[15\], leaves a remainder of \[10\] and when divided by \[16\], leaves a remainder of \[11\]
A. \[115\]
B. \[235\]
C. \[247\]
D. \[475\]

Answer 1

Hint: For solving this question, we will use a basic concept of mathematics that if a number \[n\] leaves remainder \[r\] when divided by a number \[q\] then, \[n + q - r\] will be a multiple of \[q\]. After that, we will find the L.C.M. of \[12\], \[15\] and \[16\] and solve for the correct value of the required number.

Complete step by step answer:
We have to find the least number which when divided by \[12\], leaves a remainder of \[7\], when divided by \[15\], leaves a remainder of \[10\] and when divided by \[16\], leaves a remainder of \[11\].Now, let the required number is \[n\].We know that if a number \[n\] leaves remainder \[r\] when divided by a number \[q\] then, \[n + q - r\] will be a multiple of \[q\].

And it is given that when \[n\] is divided by \[12\], it leaves a remainder of \[7\], which means \[n + 5\] will be a multiple of \[12\]. And it is given that when \[n\] is divided by \[15\], it leaves a remainder of \[10\], which means \[n + 5\] will be a multiple of \[15\]. Also, it is given that when \[n\] is divided by \[16\], it leaves a remainder of \[11\], which means \[n + 5\] will be a multiple of \[16\]. Thus, we conclude that \[n + 5\] will be a multiple of \[12\], \[15\] and \[16\].

Now, for the least value of \[n\], the value of \[n + 5\] will be the L.C.M. of \[12\], \[15\] and \[16\].
We can write \[12 = 1 \times 2 \times 2 \times 3\], \[15 = 1 \times 3 \times 5\] and \[16 = 1 \times 2 \times 2 \times 2 \times 2\].
Then, \[LCM = 1 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5\]
On solving, we get
\[ \Rightarrow LCM = 240\]
Now, we can write,
\[ \Rightarrow n + 5 = 240\]
On solving, we get
\[ \therefore n = 235\]
Therefore, the least number which when divided by \[12\], leaves a remainder of \[7\], when divided by \[15\], leaves a remainder of \[10\] and when divided by \[16\], leaves a remainder of \[11\] is \[235\].

Hence, option B is correct.

Note: To find the LCM of \[12\], \[15\] and \[16\], we have used the fundamental theorem of arithmetic in which we have first expressed the number in terms of multiplication of prime numbers. Then, we find the value of the least common multiple.Then, LCM is determined by taking the highest power of every prime number in the given number.