Question

How many three-digit numbers are multiples of neither 5 nor 7?

Answer 1

• 590

Explanation:

Total 3-digit numbers 100 to 999

• 999 - 99 = 900

3 digit multiples of 5; 100 to 995

• smallest one 100 = 5*20
• greatest one 995 = 199*5
• number 199 - 4 = 195

3 digit multiples of 7:

• smallest one 105/7 = 15
• greatest one 994/7 = 142
• number 142 - 14 = 128

Total multiples of either 5 or 7 :

• 195 + 128 = 323

This number includes multiples of both 5 and 7

We need to exclude half of the multiples of 5*7= 35 as they are counted twice:

• smallest one 105 = 35*3
• greatest one 980 = 28*35
• number 28 - 2 = 26

Half of this is 26/13 = 13

• 323 - 13 = 310

The difference of 900 and 310 is the answer:

• 900 - 310 = 590