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15 Q:
Answer: D) 1022 Explanation:
Lowest 4-digit number is 1000.LCM of 3, 4 and 5 is 60.Dividing 1000 by 60, we get the remainder 40. Thus, the lowest 4-digit number that exactly divisible by 3, 4 and 5 is 1000 + (60 - 40) = 1020. Now, add the remainder 2 that's required. Thus, the answer is 1022. Answer  Hint: To solve this question we will use the concept of LCM and AP. LCM is short of the least common multiple which is the smallest possible number which is divisible by all given numbers. AP is short for arithmetic progression whose \[{{n}^{th}}\] term is determined by the formula, \[{{a}_{n}}={{a}_{1}}+\left( n-1 \right)d\], where \[{{a}_{n}}={{n}^{th}}\]term, \[{{a}_{1}}\to \] first number, \[n\to \] number of terms & d is common difference. Complete step-by-step solution: We have to find all possible four-digit numbers divisible by 5, 12, and 18.We first factor 5 and 12 and 18.Factors of 5 are,\[5=1\times 5\]Factors of 12 are,\[12=2\times 2\times 3={{2}^{2}}\times 3\]Factors of 18 are,\[18=2\times 3\times 3={{3}^{2}}\times 2\]Now we will proceed to find LCM (Least common multiple) of 5, 12 and 18.LCM of these are given as,\[\begin{align} & 2\left| \!{\underline {\, 5,12,18 \,}} \right. \\ & 3\left| \!{\underline {\, 5,6,9 \,}} \right. \\ & 3\left| \!{\underline {\, 5,2,3 \,}} \right. \\ & 2\left| \!{\underline {\, 5,2,1 \,}} \right. \\ & 5\left| \!{\underline {\, 5,1,1 \,}} \right. \\ & \left| \!{\underline {\, 1,1,1 \,}} \right. \\ \end{align}\]So, LCM of 5, 12 and 18 = \[2\times 2\times 3\times 3\times 5\]LCM of 5, 12 and 18 = \[{{2}^{2}}\times {{3}^{2}}\times 5=180\]Because our 4 digit number must be divisible by 5, 12 and 18 therefore it should be divisible by their LCM which is 180.Hence our 4 digit number must be divisible by 180.Now the smallest 4 digit number is 1000.We need a number to be divisible by 180. Hence, we will divide 1000 by 180.\[\Rightarrow \dfrac{1000}{180}=5.55\cong 6\]Therefore the smallest 4 digit number divisible by 180 can be obtained by multiplying 6 to 180.\[\Rightarrow 180\times 6=1080\] ---------- (1)So, 1080 is the smallest four-digit number divisible by 180.Similarly, the largest 4 digit number is 9999.Dividing 9999 by 180 gives,\[\dfrac{9999}{180}\cong 55\]\[\therefore \] Highest 4 digit number that is divisible by 180 is,\[180\times 55=9900\] -------------- (2)Using the terms obtained in equation (1) and (2). We can form an AP (Arithmetic progression) of terms that are divisible by 180.Then the AP is,1080, 1260, …… 9900Here first term, \[{{a}_{1}}=1080\].Common difference, d = 1260 – 1080 = 180.And \[{{n}^{th}}\] term, \[{{a}_{n}}\] = 9900.We have to find the value of n.Formula of AP \[{{n}^{th}}\] term is given as,\[{{a}_{n}}={{a}_{1}}+\left( n-1 \right)d\]Substituting all values we get;9900 = 1080 + (n - 1)180Subtracting 1080 both sides we get,$(n - 1)180 = 9900 – 1080$\[\Rightarrow (n - 1)180 = 8820 \]Dividing by 180 we get,\[\begin{align} & \left( n-1 \right)=\dfrac{8820}{180} \\ & \Rightarrow \left( n-1 \right)=49 \\ \end{align}\]Adding 1 on both sides,\[\Rightarrow \] $n = 49 + 1 = 50$Therefore, value of $n = 50.$Hence the total number of 4 digit numbers which are divisible by 5, 12, and 18 are 50, which is an option (d). Note: Approximating the largest and smallest possible four-digit number divisible by 180 is correct because any two consecutive numbers divisible by 180 have at least a difference of 180 in between. Therefore approximation works well in this case. Also, any number divisible by 180 (LCM of 5, 12, and 18) is also divisible by one or all of 5, 12, and 18. So this method works well. The UPSC Examinations 2021 Reserved List has been released. 63 candidates have been recommended by the commission to fill in the remaining posts. With reference to the 2022 exam cycle, The Union Public Service Commission (UPSC) examination was conducted on the 16th, 17th, 18th, 24th, and 25th of September 2022. This is one of the most coveted jobs in India. The candidates are required to go through a 3 stage selection process - Prelims, Main and Interview. The marks of the main examination and interview will be taken into consideration while preparing the final merit list. The candidates must go through the UPSC Civil Service mains strategy to have an edge over others. |
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