Question 12 of 18intermediate🔧 ApplyNumerical3 marks

A cowherd took all his cows to graze in the fields. The cows came to a crossing with 3 gates. An equal number of cows passed through each gate. Later at another crossing with 5 gates again an equal number of cows passed through each gate. The same happened at the third crossing with 7 gates. If the cowherd had less than 200 cows, how many cows did he have?

Correct Answer

Given: An equal number of cows passed through each gate at crossings with 3, 5, and 7 gates.

Step 1: The number must be divisible by 3, 5, and 7.

Step 2: Find LCM(3, 5, 7) = 3×53 \times 5 × 7 = 105 (since they are co-prime).

Step 3: Find multiples of 105 less than 200: only 105.

Answer: The cowherd had 105 cows.

Exercise: Figure it Out (Section 6) | Q: 14 | (Page Page 17)
For More Understanding

Explanation

This is an LCM problem. The number of cows must be a common multiple of 3, 5, and 7. Since 3, 5, and 7 are co-prime (pairwise), their LCM is simply their product: 3×53 \times 5 × 7 = 105. The next common multiple would be 210, which exceeds 200. Therefore, the cowherd had 105 cows.

Solution Steps

  1. Step 1: The number must be divisible by 3, 5, and 7.

  2. Step 2: Find LCM(3, 5, 7) = 3×53 \times 5 × 7 = 105 (since they are co-prime).

  3. Step 3: Find multiples of 105 less than 200: only 105.

  4. Step 4: The cowherd had 105 cows.

PreviousNext