Question

there are certain number of soldiers in the field in the soldiers are arranged in rows of 8 of 15 and 20 soldier is left out if the soldiers are in the roles of 9 or 13, soldiers only are left out find the number of soldiers in the field ​

Answer 1

Answer:

121

Step-by-step explanation:

take the lcm of 8,15,20 = 120. since 1 soldier is left then add 1 to the lcm = 121. divide and check this answer by 9 or 13 if the condition is true then 121 is the answer

Answer 2

• The number of soldiers in the field are equal to 121 .

Correct Question :- There are certain number of soldiers in the field . If the soldiers are arranged in rows of 8 or 15 or 20, one soldier is left out . If the soldiers are arranged in the rows of 9 or 13, only four soldiers are left out . Find the number of soldiers in the field ?

Concept used :-

• The least number which when divided by x, y and z leaves the same remainder ‘r’ each case is equal to = LCM(x, y and z) + r .

Solution :-

Case 1) :- When soldiers are arranged in rows of 8, 15 or 20 .

→ Possible number of soldiers in the field = 8, 15 or 20 .

→ Soldiers left out = 1

So,

→ Total number of soldiers in the field = LCM(8, 15, 20)•m + 1 { where m is constant. }

Finding LCM of 8, 15 and 20 by prime factorisation method :-

→ 8 = 2 × 2 × 2

→ 15 = 3 × 5

→ 20 = 2 × 2 × 5

then,

→ LCM(8, 15, 20) = 2 × 2 × 2 × 3 × 5 = 120

therefore,

→ Total number of soldiers in the field = LCM(8, 15, 20)•m + 1

→ Total number of soldiers in the field = (120m + 1) ---------- Equation (1)

Case 2) :- When soldiers are arranged in rows of 9 or 13 .

→ Possible number of soldiers in the field = 9 or 13.

→ Soldiers left out = 4

So,

→ Total number of soldiers in the field = LCM(9, 13)•n + 4 { where n is constant. }

Finding LCM of 9 and 13 by prime factorisation method :-

→ 9 = 3 × 3

→ 13 = 1 × 13

then,

→ LCM(9, 13) = 3 × 3 × 13 = 117

therefore,

→ Total number of soldiers in the field = LCM(9, 13)•n + 4

→ Total number of soldiers in the field = (117n + 4) ---------- Equation (2)

since total number of soldiers in field are equal in both cases . Therefore,

→ Equation (1) = Equation (2)

→ (120m + 1) = (117n + 4)

→ 120m - 117n = 4 - 1

→ 120m - 117n = 3

putting m = n = 1 we get,

→ 120 × 1 - 117 × 1 = 3

→ 120 - 117 = 3

→ 3 = 3

as we can see that, the above equation is satisfied when m = n = 1 .

hence,

→ Total number of soldiers in the field = 120m + 1 = 120 × 1 + 1 = 120 + 1 = 121 (Ans.)

Or,

→ Total number of soldiers in the field = 117n + 4 = 117 × 1 + 4 = 117 + 4 = 121 (Ans.)

Learn more :-

वह छोटी से छोटी संख्या बताईये जिसमे 7,9,11 से भाग देने पर 1,2,3 शेष बचे

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