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It is not necessary to use the Sears store branded card (Citi is a advertising partner) to participate in the Shop Your Way rewards program. Actually it would be more beneficial to use a traditional bank credit card that offers rewards because that would allow you to double your rewards.

Most traditional rewards credit cards offer at least one mile or point for every dollar spent.  So you would be earning at least 2% in rewards by using both programs. The Shop Your Way Program offers reward points that can be used towards future store purchases. Bonus points are offered in rotating categories weekly. If you are redeeming more than $10 in points you will need to enter your PIN.

The Sears Card® itself does not earn rewards.  It is tied to the Shop Your Way rewards program, which has no annual fee and is free to join. You don't even need a rewards card. Just sign up and provide your phone number at checkout and discounts and points are applied automatically no matter what credit card you use.