Question

Find the greatest four-digit number which when divided by $20$, $30$, $35$ and $45$ leaves remainder $12$ in each case.

• A. $8751$
• B. $8242$
• C. $8832$
• D. $9892$

Answer 1

The correct option is D. $8832$

Prime factor of $20,30,35$ and $45$

$20=2^2\times 5$

$30=2\times 3\times 5$

$35=5\times 7$

$45=3^2\times 5$

LCM=product of each prime factor of highest power

LCM=$2^2\times 3^2\times 5\times 7=1260$

Greatest four digit number=$9999$

Greatest four digit number divisible by all given numbers=$9999-$remainder when 9999 is divided by LCM of given numbers

Greatest four digit number divisible by given numbers=$9999-1179=8820$

Given that,

Required number when divided by $20,30,35,45$ leaves remainder $12$.

Therefore ,required number=$8820+12=8832$

Hence greatest four digit number when divided by $20,30,35$ and $45$ is $8832$.