Question

A basket has at most 150 applies in it. Then the number of apples in the basket if the number of apples satisfy the following conditions If counted in pairs, one remains; If counted in threes, two remains; If counted in fourss, three remains; If counted in fives, four remains; If counted in sixes, five remains; If counted in sevens, nothing remains; (A) 116 (B) 117 (C) 118 (D) 119

Answer 1

Number of apple = 119

Explanation:

Given

`Max_(Apple) = 150`

Required

Find the actual number of apples;

When counted in any numbers, the number being counted is the divisor;

Take for instance, when counted in pairs, the divisor is 2

When counted in threes, the divisor is 3

and so on...

So; from the question, we understand that; when the apple is counted in 2s,3s,4s,5s, and 6s; what's left is always one less than the divisor

This means that the number of apple must satisfy the following conditions

• Less than 150
• 1 less than the Common Multiple of 2,3,..,6
• Divisible by 7

The common multiple of 2,3,..6 are 60, 120 ,180,240.....

Recall that, these multiples must satisfy the three conditions above;

1. Less than 150: Only 60 and 120 satisfy condition.

At this point, we'll discard other numbers (180,240,300.....)

2. 1 less than the Common Multiple of 2,3,..,6:

1 less than 60: 60 - 1 = 59

1 less than 120: 120 - 1 = 119

3. Divisible by 7

At this point, we'll divide 59 and 119 by 7 to see if they satisfy this condition;

`(59)/(7) = 8.42857142857`

`(119)/(7)=17`

The results of these division shows that 59 is not divisible by 7;

Hence, the actual number of apple is 119

Number of apple = 119