Question

The teacher wrote the three-digit number with unit digit 8 he imagined on the board. Three students made two assumptions about the given
number.
Student 1: All digits are even and it has an even number of different prime factors.
Student 2: It is a multiple of 9 and it is a perfect square.
Student 3: it is less than 400 and it is 13 times greater than one perfect square.
Which number did the teacher imagine if each student guessed just one assumption?​

Answer 1

Solution :-

Student 1) :- All digits are even and it has an even number of different prime factors.

• 2 has even number of prime factors => 1 and 2 .
• 4 has odd number of factors = 3 .
• 6 has even number of factors = 4 .
• 8 has even number of factors = 4 .

so, Possible 3 digits numbers with 2,6 and 8 are :-

• 268, 286, 628, 682, 826, 862 .

Student 2) :-It is a multiple of 9 and it is a perfect square.

• Perfect square 3 digits numbers are :- 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900 and 961 .
• Multiple of 9 out of these are :- 144, 225, 324, 441, 576, 729 and 900 .

Student 3) :- it is less than 400 and it is 13 times greater than one perfect square.

Possible numbers are :-

• (3)² + 9 * 13 = 126
• (4)² + 16 * 13 = 224
• (5)² + 25 * 13 = 350 .

now, we have given that, The teacher wrote the three-digit number with unit digit 8 he imagined on the board.

therefore, student (2) and student (3) assumptions are wrong .

Student (1) assumption is correct , All digits are even and it has an even number of different prime factors and required three digits numbers ends with 8 in this case are either 268 or 628 .

Learn more :-

let a and b positive integers such that 90 less than a+b less than 99 and 0.9 less than a/b less than 0.91. Find ab/46

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