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# help counting

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Andrew chooses a number from 1 to 100, and Mary also chooses a number from 1 to 100. (They may choose the same number.) It turns out that the product of their numbers is a multiple of 10. In how many ways could Andrew and Mary have chosen their numbers?

Jun 17, 2021

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For product 10 = {(1,10), (2,5)}                           = 2 pairs

For product 20 = {(1,20), (2,10), (4,5)}               = 3 pairs

For product 30 = {(1,30), (2,15), (3,10), (5,6)}    = 4 pairs

For product 40 = {(1,40), (2,20), (4,10), (5,8)}    = 4 pairs

For product 50 = {(1,50), (2,25), (5,10)}              =3 pairs

For product 60 = {(1,60), (2,30), (3,20), (4,15), (5,12), (6,10)}   = 6 pairs

For product 70 = {(1,70), (2,35), (5,14), (7,10)}                          = 4 pairs

For product 80 = {(1,80), (2,40), (4,20), (5,16), (8,10)}              = 5 pairs

For product 90 = {(1,90), (2,45), (3,30), (5,18), (6,15), (9,10)}   = 6 pairs

For product 100 = {(1,100), (2,50), (4,25), (5,20), (10,10)}        = 5 pairs

⇒ No. of ways = 2 × (2 + 3 + 4 +  4  + 3 + 6 + 4 + 5 + 6 + 5)

= 2 × 42

= 84

Jun 18, 2021