Find the least multiple of 23 which when divided by 18,21 and 24 leaves remainder 7,10 and 13 respectively

Miranda Johns

Miranda Johns

Answered question

2023-01-25

Find the least multiple of 23 which when divided by 18,21 and 24 leaves remainder 7,10 and 13 respectively

Answer & Explanation

Ezekiel Robbins

Ezekiel Robbins

Beginner2023-01-26Added 8 answers

Given information
When a number is divided by 18, a residue of 7 is left over.
When the same number is divided by 21, it leaves a remainder of 10.
When the same number is divided by 24, it leaves a remainder of 13.
Observing the constant difference of 11 between divisor and remainder:
We may infer that when 11 is deducted from the LCM of 18, 21, and 24, it will result in a remainder of 7, 10, or 13 when divided by each of the three numbers in the LCM.
When 11 is subtracted from any multiple of LCM, it will give the same result.
So,
18=2×3×3
21=3×7
24=2×2×2×3
∴ LCM of 18,21 and 24 = 2 3 × 3 2 × 7 = 504
Now, any number of the form (504n–11) will give the required remainders, when n is an integer.
But, this number also has to be divisible by 23.
Using trial and error method, for n=6:
We find that, 504n–11=504×6–11=3024–11=3013
Hence, the required number which satisfies all the conditions is 3013.

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