Find the smallest number which when divided by 24 and 36 leaves no remainder

Answer

Hint: Observe that 21 =24-3, 33 = 36 – 3 and 45 = 48 -3.Hence if a be the smallest number which on division by 21, 33 and 45 leaves remainder 0, then a-3 is the smallest number which on division by 24, 36 and 45 leaves remainder 21, 33 and 45.Complete step-by-step answer:Since 21 =24-3, 33 = 36 – 3 and 45 = 48 -3, if a is the smallest number which on division by 21, 33 and 45 leaves remainder 0, then a-3 is the smallest number which on division by 24, 36 and 45 leaves remainder 21, 33 and 45.Let L be the LCM of 24,36 and 48. Hence from the definition of L, we have 24|L, 36|L and 48|L {a|b is read as a divides b}. Also let m be a natural number such that 24|M, 36|M and 48|M then we have $L\le M${Because M is a common multiple of 24,36 and 48 and L is least common multiple.}Hence, we have a = L.Now we know that $\begin{align}  & 24={{2}^{3}}\times 3 \\  & 36={{2}^{2}}\times {{3}^{2}} \\  & 48={{2}^{4}}\times 3 \\ \end{align}$Hence, we have,$L={{2}^{4}}\times {{3}^{2}}=144$Hence a = 144.Hence the smallest number which divides by 24,36 and 48 leaves remains 21, 33 and 45 respectively is a-3 = 144-3 = 141.Note: [1] We will prove that if a is the smallest number which on division by 21, 33 and 45 leaves remainder 0, then a-3 is the smallest number which on division by 24, 36 and 45 leaves remainder 21, 33 and 45.Proof:As shown above a = L = LCM (24,36,45).Since 24>3,36>3 and 45>3, we have L>3. So, a-3>0.Let a-3 is not the smallest number with the above properties and let m be such a number.

So, we have m

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150 Questions 600 Marks 135 Mins

Given:

Divisor = 24, 32 and 36

Reminder = 19, 27 and 31

Calculation:

Difference between divisor and reminder

24 - 19 = 5

32 - 27 = 5

31 - 31 = 5

The difference between divisors and reminders is the same in all

According to the question, we have

The LCM of (24, 32 and 36) = 288

Now,

The smallest number = LCM - 5

⇒ 288 - 5

⇒ 283

∴ The smallest number is 283, divisible by 24, 32 and 36 and leaves the remainder 19, 27 and 31 respectively.

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